System and Method for Estimating Permeability of a Bioturbated Reservoir

The present application claims benefit of priority to U.S. Provisional Application No. 63/670,523 having a filing date of Jul. 12, 2024 which is incorporated herein by reference in its entirety.

Aspects of the present disclosure are described in “Digital rock modeling to quantify scale dependence of petrophysical measurements in burrowed reservoir rocks: An example using Thalassinoides”, published in Marine and Petroleum Geology, Volume 155, 106412, which is incorporated herein by reference in its entirety.

Support provided by the College of Petroleum Engineering and Geosciences in King Fahd University of Petroleum and Minerals, Saudi Arabia, under research startup grant SF19031 is gratefully acknowledged.

The present disclosure is directed towards digital rock modelling, and more particularly, directed towards a system and a method for estimating a permeability of a bioturbated reservoir.

The “background” description provided herein is to present the context of the disclosure generally. Work of the presently named inventors, to the extent it is described in this background section, as well as aspects of the description that may not otherwise qualify as prior art at the time of filing, are neither expressly nor impliedly admitted as prior art against the present invention. Permeability of subsurface carbonate reservoirs is an important parameter for formation evaluation and fluid flow simulation. Yet, permeability cannot be measured directly from rocks of subsurface reservoirs using conventional well logs. However, permeability may be inferred from well testing and production or may be modelled, with varying degrees of success, from well logs such as porosity and NMR logs. Reservoir permeability has been estimated through laboratory measurements from cored reservoir rocks. Such laboratory measurements have been used to calibrate well logs to calculate a permeability log that is eventually used in fluid flow simulation.

The rock samples used for such laboratory measurements have core plugs diameters in the range of 1 inch to 1.5 inch and conventional reservoir core diameter of about 4 inch. These diameters are small compared to the actual scale of the pore systems in many carbonate reservoirs. In some cases, the dimensions of the samples analyzed are not large enough to capture the dimensions that include the representative elementary volume (REV). REV refers to the smallest volume of a sample with correct dimensions from which a measured parameter is size-independent. Measured permeabilities on such small samples are unlikely to represent the actual permeability in reservoirs, because of the large scale of the heterogeneity in pore system of the reservoir. It has been demonstrated that use of such permeability measurements leads to errors when predicting reservoir performance. One source of such large-scale permeability heterogeneity is induced by burrows in carbonate reservoir rocks. In carbonate strata, burrows may enhance the porosity and permeability of a tight rock matrix, if the burrows remain open or are filled with sediment with porosity and permeability higher than the matrix. An example is carbonate reservoir with Thalassinoides, which refers to a branched burrow with a boxwork pattern and horizontal, as well as vertical penetration. In many carbonate reservoirs, Thalassinoides infills may be dolomitized and host intercrystalline porosity, making them considerably more porous and permeable than a surrounding limestone matrix. In some carbonate reservoirs, Thalassinoides are infilled with grain-dominated sediments that include interparticle porosity with substantially more porosity and permeability relative to surrounding muddy carbonate. Further, Thalassinoides can be open or partially open, when this occurs, the burrow porosity of Thalassinoides may result in flow zones that are extremely permeable. In cases where Thalassinoides enhances the porosity and permeability of carbonate reservoirs, burrow connectivity controls the permeability, because the connected Thalassinoides network provides permeability passageways in otherwise less permeable host rock matrix. Capturing the full connectivity of Thalassinoides networks is not guaranteed. Small sample sizes may, or may not, be suitable to represent the permeability of Thalassinoides bearing carbonate reservoirs. Determining what minimum sample size is needed to represent permeability and porosity requires systematically analyzing a variety of burrow sizes, burrow percentages, and sample dimensions. Getting such a variety from natural samples may be highly challenging. Therefore, in order to overcome above stated challenges, a need arise for an efficient and accurate digital modeling approach as an efficient alternative.

Hence, it is one object of the present disclosure to provide a system for estimating a permeability of a bioturbated reservoir and a method thereof, that may circumvent the aforementioned drawbacks.

In an exemplary embodiment, a method of estimating a permeability of a bioturbated reservoir based on a Thalassinoides connectivity is described. The method includes generating a plurality of geocellular models from a Thalassinoides morphology, converting the plurality of geocellular models to a plurality of training images each having a host rock matrix and Thalassinoides burrows. The method further includes measuring a plurality of statistical parameters from the plurality of training images to obtain a width of a Thalassinoides shaft, creating a plurality of samples each having a burrow percentage, a burrow size, and a sample cross section from the plurality of geocellular models. Further, the method includes determining a largest connected burrow volume (LCBV) of each sample of the plurality of samples based on the Thalassinoides burrows to obtain a burrow connectivity and computing the Thalassinoides connectivity based on the burrow connectivity to thereby estimate the permeability of the bioturbated reservoir.

3 In some embodiments, each geocellular model of the plurality of geocellular models includes a three-dimensional multipoint statistics (3DMPS) model having a three-dimensional volume of about 1 m.

In some embodiments, the plurality of geocellular models includes 18 3DMPS models.

In some embodiments, each 3DMPS model of the 18 3DMPS models is constructed based on an Eltom method.

2 2 In some embodiments, the creating further includes extracting a plurality of columnar samples from each geocellular model of the plurality of geocellular models, extracting a plurality of subsamples each having a column cross section from each columnar sample of the plurality of columnar samples, and combining the plurality of subsamples to obtain the plurality of samples. An area of the column cross section is between 25 cmand 900 cm.

2 2 2 2 2 2 In some embodiments, the plurality of subsamples includes 6 subsamples and wherein the area of the column cross section of the plurality of subsamples is 25 cm, 100 cm, 225 cm, 400 cm, 625 cm, or 900 cm.

In some embodiments, the burrow percentage of each sample of the plurality of samples is selected from the group consisting of 20%, 50%, and 75%.

In some embodiments, the burrow size of each sample of the plurality of samples is between 2.6 cm and 9 cm.

In some embodiments, the determining further includes determining the LCBV of each sample of the plurality of samples based on an Eltom method, determining whether the LCBV of each sample of the plurality of samples is connected across from a top to a bottom of each sample of the plurality of samples, indicating, when the LCBV is connected across from the top to the bottom, the LCBV as a connected burrow, and measuring a length and a position of the LCBV to determine the burrow connectivity.

In some embodiments, the computing further includes dividing the plurality of samples into a training set and a validation set, running a logistic regression analysis with the training set and the burrow connectivity to obtain a logistic regression result. The computing further includes validating the logistic regression result with the validation set and the burrow connectivity to obtain a probability equation, and computing the Thalassinoides connectivity based on the probability equation.

In another exemplary embodiment, a system for estimating a permeability of bioturbated reservoirs represented by a Thalassinoides connectivity is described. The system includes a processor configured to execute a program instruction, and a memory having the program instruction, where the memory is connected to the processor. The system further includes an input device connected to the processor and configured to receive a plurality of computed tomography (CT) scan images each having a Thalassinoides morphology, and a display device configured to display the Thalassinoides connectivity. The program instruction includes generating a plurality of geocellular models from the Thalassinoides morphology of the plurality of CT scan images. The program instruction further includes converting the plurality of geocellular models to a plurality of training images each having a host rock matrix and Thalassinoides burrows, measuring a plurality of statistical parameters from the plurality of training images to obtain a width of a Thalassinoides shaft, creating a plurality of samples each having a burrow percentage, a burrow size, and a sample cross section from the plurality of geocellular models. The program instruction further determines a largest connected burrow volume (LCBV) of each sample of the plurality of samples based on the Thalassinoides burrows to obtain a burrow connectivity, and computes the Thalassinoides connectivity based on the burrow connectivity to thereby estimate the permeability of bioturbated reservoirs.

3 In some embodiments, each geocellular model of the plurality of geocellular models includes a three-dimensional multipoint statistics (3DMPS) model having a three-dimensional volume of about 1 m.

In some embodiments, the plurality of geocellular models includes 18 3DMPS models.

In some embodiments, each 3DMPS model of the 18 3DMPS models is constructed based on an Eltom method.

2 2 In some embodiments, the creating further includes extracting a plurality of columnar samples from each geocellular model of the plurality of geocellular models, extracting a plurality of subsamples each having a column cross section from each columnar sample of the plurality of columnar samples, and combining the plurality of subsamples to obtain the plurality of samples. An area of the column cross section is between 25 cmand 900 cm.

2 2 2 2 2 2 In some embodiments, the plurality of subsample includes 6 subsamples and where the area of the column cross section of the plurality of subsamples is 25 cm, 100 cm, 225 cm, 400 cm, 625 cm, or 900 cm.

In some embodiments, the burrow percentage of each sample of the plurality of samples is selected from the group consisting of 20%, 50%, and 75%.

In some embodiments, the burrow size of each sample of the plurality of samples is between 2.6 cm and 9 cm.

In some embodiments, the determining further includes determining the LCBV of each sample of the plurality of samples based on an Eltom method, determining whether the LCBV of each sample of the plurality of samples is connected across from a top to a bottom of each sample of the plurality of samples, indicating, when the LCBV is connected across from the top to the bottom, the LCBV as a connected burrow, and measuring a length and a position of the LCBV to determine the burrow connectivity.

In some embodiments, the computing further includes dividing the plurality of samples into a training set and a validation set, running a logistic regression analysis with the training set and the burrow connectivity to obtain a logistic regression result, validating the logistic regression result with the validation set and the burrow connectivity to obtain a probability equation, and computing the Thalassinoides connectivity based on the probability equation.

The foregoing general description of the illustrative present disclosure and the following detailed description thereof are merely exemplary aspects of the teachings of this disclosure and are not restrictive.

In the drawings, reference numerals designate identical or corresponding parts throughout the several views. Further, as used herein, the words “a,” “an,” and the like generally carry a meaning of “one or more,” unless stated otherwise.

Furthermore, the terms “approximately,” “approximate,” “about,” and similar terms generally refer to ranges that include the identified value within a margin of 20%, 10%, or preferably 5%, and any values therebetween.

Aspects of the present disclosure are directed towards a system and a method for estimating permeability of bioturbated reservoirs represented by a Thalassinoides connectivity. Thalassinoides is a trace fossil including branching burrows in sedimentary rock, attributed to ancient marine or freshwater crustaceans known as ghost shrimp. The burrows often display characteristics of Y-shaped and U-shaped branching pattern, indicating an activity of crustaceans as they excavated through sedimentary rock formation. Since burrow connectivity have shown that burrow morphology, burrow abundance, and burrow size are factors impacting a sampling scheme required for representing permeability of bioturbated reservoir, the present disclosure discloses use of multipoint statistics modelling and applies it to Thalassinoides. The present disclosure aims to simulate a range of burrow abundances and sizes, and subsequently interrogate the outcome to develop a statistical model that allows sampling strategies to be designed.

1 FIG. 100 100 100 102 104 102 104 104 102 106 102 106 104 106 102 102 3 3 6 Referring to, a schematic block diagram of a computing environment including a systemis illustrated, according to certain embodiments. In particular, the systemis designed for estimating a permeability of bioturbated reservoirs represented by a Thalassinoides connectivity. In general, bioturbated reservoirs refer to geological formations where the sedimentary layers have been extensively reworked by physical activity of organisms. The bioturbation process may alter the original bedding and sediment structures, leading to a more homogenized and often complex arrangement of grains and particles of the geological formations. The activity of such organisms, typically invertebrates like worms and bivalves, can significantly impact properties of the reservoirs, such as porosity and permeability, which are crucial for the storage and flow of fluids like oil, gas, and water. In some embodiments, the systemincludes a processorand a memory. The processoris configured to execute program instruction stored in the memory. In other words, the memoryhaving program instruction is connected to the processerelectronically. In some embodiments, an input deviceis connected to the processorand the input deviceis configured to receive a plurality of computed tomography (CT) scan images. Further, each CT scan image has a Thalassinoides morphology. In an embodiment, the program instruction, as stored in the memory, is a set of digital instructions which includes generating a plurality of geocellular models from the Thalassinoides morphology of the plurality of CT scan images received by the input device. In particular, a large diameter core of a particular bioturbated reservoir underwent CT scans in order to generate the plurality of CT scan images. The processorexecutes the program instructions to create a high-resolution geocellular model. In some embodiments, each of the plurality of geocellular models includes a three-dimensional multipoint statistics (3DMPS) model having a three-dimensional volume of about 1 cubic meter (m). In general, a geocellular model is a three-dimensional representation of a geological space of bioturbated reservoirs, used for characterization and simulation. Geocellular models incorporates a plurality of geological, geophysical, and petrophysical data to create a detailed depiction of subsurface structures. The geocellular model includes cellular grids that align with major structural components like faults and horizons, allowing for greater accuracy in simulating fluid flow and reservoir potential. Geocellular models are crucial for making informed decisions in field development and operational planning, as they help understand how complex reservoir attributes influence fluid dynamics. Hence, the geocellular models created by the processorupon execution of program instructions, and specifically, upon processing CT scan images allows for greater accuracy in determining the permeability of the bioturbated reservoir. Further, the plurality of geocellular models includes 18 3DMPS models. In an aspect, each 3DMPS model of the 18 3DMPS models is constructed based on an Eltom method. The “Eltom method” refers to creating 3DMPS models having three variables: burrow morphology (three end-member burrow morphologies boxwork, vertical and horizontal), bioturbation intensity expressed as burrow volume percentage (25 intensities, ranging from 2% to 50%, for example), and matrix properties held consistent for all models. These variables were simulated in the high-resolution geocellular models (e.g. 1 mmodel having 8×10cells, resulting in 75 models).

3 2 2 2 2 2 2 2 In some embodiments, the program instruction is encoded in such a way that they convert the plurality of geocellular models to a plurality of training images, each image having a host rock matric and Thalassinoides burrows. In some embodiments, the program instruction further includes measuring a plurality of statistical parameters from the plurality of training images to obtain a width of a Thalassinoides shaft. The program instruction creates a plurality of samples, each have a burrow percentage (BP), a burrow size (BS), and a sample cross section from the plurality of geocellular models. In other words, the plurality of training images is used to generate modelled Thalassinoides with a range of bioturbation intensity expressed as the BP and the BS. In an example, six Thalassinoides burrow sizes varying from 2.6 cm to 9 cm were modelled by varying a cell size of the 1 m3D volume of the plurality of geocellular models. In an embodiment, creating the plurality of samples further includes extracting a plurality of columnar samples from each geocellular model of the plurality of geocellular models. In addition, creating the plurality of samples further includes extracting a plurality of subsamples each having a column cross section from each columnar sample of the plurality of columnar samples, where an area of the column cross section is between 25 cmand 900 cm. In an example of the present disclosure, each 3DMPS model with a single BP and BS was extracted into 30 columnar samples, thus a total of 18 3DMPS model (as stated above) were converted into 540 columnar samples to evaluate a range of BP, BS, and sample cross section. The BP of each sample of the plurality of samples is selected from the group consisting of 20%, 50%, and 75%. As described above, the BS of each sample of the plurality of samples is between 2.6 cm and 9 cm. In other words, the plurality of subsamples was combined in order to obtain the plurality of subsamples. Examination of the 540 samples may help in determining the cross-sectional area needed to represent permeability of the reservoir, based on which the samples were modelled. Further, the plurality of subsamples includes 6 subsamples. The area of the column cross section of the plurality of subsamples is 25 cm, 225 cm, 400 cm, 625 cm, or 900 cm. In an example, the burrow connectivity (permeability) was analysed in a computing software provided by Petrel™.

104 100 104 104 The program instruction stored in the memoryof the systemfurther includes determining a largest connected burrow volume (LCBV) of each sample of the plurality of samples. The LCBV is determined based on the Thalassinoides burrows to obtain a burrow connectivity. In some embodiments, determining the LCBV further includes determining the LCBV of each sample of the plurality of samples based on the Eltom method. The program instruction stored in the memorydetermines whether the LCBV of each sample of the plurality of samples is connected across from a top to a bottom of each sample of the plurality of samples, and further indicates, when the LCBV is connected across from the top to the bottom, the LCBV as a connected burrow. In other words, the LCBV may be considered a proxy for burrow connectivity, as such, if the LCBV touches an upper face and a lower face of the plurality of samples (top and bottom of each sample), then the 3DMPS model has vertical connected burrow and subsequently the bioturbated reservoirs have high permeability. In addition, the program instruction measures a length and a position of the LCBV to determine the burrow connectivity. Furthermore, the program instruction stored in the memoryincludes computing the Thalassinoides connectivity based on the burrow connectivity to thereby estimate the permeability of the bioturbated reservoirs. The computing further includes dividing the plurality of samples into a training set and a validation set. In an example, each of the 540 samples were randomly divided into two subsets (training set and validation set). A total of 432 samples were used as the training set, running a logistic regression analysis with the training set and the burrow connectivity to obtain a logistic regression result for regression modelling. Further, a total of 108 samples out of the 540 samples were used as external validation samples, validating the logistic regression result with the validation set and the burrow connectivity to obtain a probability equation. The validation is conducted by comparing a predicted binary class to their actual class, of the Thalassinoides connectivity against an isolated burrow volume from the logistic regression model. Such comparison may assist in evaluating sensitivity and specificity of the logistic regression results. In some embodiments, the Thalassinoides connectivity is computed based on the aforementioned probability equation. A mathematical model of the probability equation and other specific details of logistic regression modelling are provided in the subsequent paragraph(s) in the ‘examples’ section.

100 108 108 108 102 Further, the systemincludes a display deviceconfigured to display the Thalassinoides connectivity. In particular, the display devicemay refer to a graphic display, a digital display, a touch screen display, a dot-matrix display, a LED display, an LCD display, or a combination thereof. In some embodiments, the display deviceis configured to receive a set of data from the processorto display the set of data by converting it into a readable medium.

2 FIG. 200 200 200 200 200 102 100 Referring to, a schematic flow chart of a methodfor estimating the permeability of the bioturbated reservoir based on the Thalassinoides connectivity is illustrated, according to certain embodiments. The order in which the methodis described is not intended to be construed as a limitation, and any number of the described method steps can be combined in any order to implement the method. Additionally, individual steps may be removed or skipped from the methodwithout departing from the spirit and scope of the present disclosure. In particular, the methoddescribes a flow and order of implementation of the processes included in the processorof the system. The method steps highlight major processes involved in the estimation of the permeability of bioturbated reservoirs. More detailed explanation of implementation of the method steps is provided in the ‘examples’ section of the present disclosure for sake of brevity in explanation.

202 200 100 202 100 202 100 3 At a step, the methodincludes generating the plurality of geocellular models from the Thalassinoides morphology. As described with respect to the system, each geocellular model of the plurality of geocellular models includes the 3DMPS model having 3D volume of about 1 m, further, other details pertaining to the plurality of models generated at the stepremains similar to the plurality of geocellular models of the system. The stepdetails the process of generating the plurality of geocellular models, this helps in regard to estimating the permeability of the bioturbated reservoirs more accurately. Further, the plurality of geocellular models may be used to train the systemin order to produce consistent results.

204 200 204 At a step, the methodincludes converting the plurality of geocellular models to a plurality of training images each having a host rock matrix and Thalassinoides burrows. The host rock matrix and the Thalassinoides burrows are collectively known as rock fabrics. The rock fabrics and associated parameters are crucial for generation of accurate digital models of the Thalassinoides with a range of bioturbation intensity. The training images of stepmay further be used to produce algorithms for sample dimensions that may represent the permeability of a particular reservoir correctly. Thus, the plurality of geocellular models are modelled with high accuracy and meticulous attention to detail in order to produce the training images with high accuracy.

206 200 204 204 206 206 At step, the methodincludes measuring a plurality of statistical parameters from the plurality of training images generated at step, to obtain a width of a Thalassinoides shaft. The Thalassinoides shafts and their corresponding sizes may indicate a source of the Thalassinoides. Further, in an example, JMicroVision open source software was used for the analysis of the plurality of images generated at step, for accurate measurement of the Thalassinoides shaft at step. In some aspects, the Thalassinoides shaft and corresponding dimensions may be used as a proxy for burrow size. Furthermore, statistical parameters such as, mean, median, minima, maxima, and standard deviation may be used to conduct the measuring of the width of the Thalassinoides shaft as described in step.

208 200 208 200 210 200 2 2 At step, the methodincludes creating the plurality of samples each having the BP, the BS, and the sample cross section from the plurality of geocellular models. At step, the methodfurther includes extracting the plurality of columnar samples from each geocellular model of the plurality of models. The column cross section of the plurality of columnar samples is between 25 cmto 900 cm. As such, 540 columnar samples are analyzed to obtain the burrow connectivity. Further, at step, the methodincludes determining the LCBV of each sample of the plurality of samples based on the Thalassinoides burrows to obtain the burrow connectivity. In some aspects, the BP, the BS, and the sample cross section are parameters that may be used to determine the LCBV. The LCBV may be determined from top of the sample to the bottom of the sample, or, from one side of the sample to another side of the sample. If the LCBV connected the top of the sample to the bottom of the sample, then the sample may be marked as permeable, and further may percolate a fluid from the top to the bottom of the sample. In case the LCBV does not touch the top and the bottom surfaces of the sample, then the burrow volumes may not provide permeability pathways from the bottom to the top of a measured sample.

212 200 200 100 200 202 212 At step, the methodincludes computing the Thalassinoides connectivity based on the burrow connectivity to thereby estimate the permeability of the bioturbated reservoir. The methodfurther includes dividing the plurality of samples into training set and validation set. The logistic regression modelling, as described above with respect to systemis then applied in order to generate a probability equation to determine the permeability of bioturbated reservoirs. Further, the burrow connectivity approach based on the LCBV may be applied to any trace fossil morphology. A machine learning model may further be trained on the basis of the methodand subsequent stepsto. The machine learning model may determine patterns of relationship between the BP, the BS, sample cross section, sample length, burrow morphology, and burrow connectivity.

100 200 The following examples demonstrate the systemfor estimating the permeability of the bioturbated reservoir based on the Thalassinoides connectivity, and the methodthereof. The examples are provided solely for illustration and are not to be construed as limitations of the present disclosure, as many variations thereof are possible without departing from the spirit and scope of the present disclosure.

3 3 3 3 3 3 3 3 2 3 FIG.A 3 FIG.A In accordance with the present disclosure, 18 MPS three-dimensional (3D) simulation scenarios of Thalassinoides were modelled in a 3D grid with a volume of about 1 cubic meter (m), as shown in. The MPS models were constructed in Petrel™ 2020 using the methods described by Eltom and coworkers in various studies from 2019 to 2021. The workflow starts with capturing Thalassinoides morphology in a high-resolution geocellular model with a 3D volume of about 1 mand converting the geocellular model to a training image. The training image includes two rock fabrics: the host rock matrix, and the Thalassinoides burrows. The training image is used to generate modelled Thalassinoides with a range of bioturbation intensity, expressed as burrow percentage (BP) and burrow size (BS), as shown in. Each one of the 18 scenarios of Thalassinoides was modelled in a 1 m3D volume representing three BPs of about 25%, 50%, and 75%. The above specified BPs were selected as previous experimentation had indicated these BPs as a likely range of BP covering natural systems, yet still, with 100% probability, that burrows connect across the sides of the 1 m3D volume. This allows for evaluation of the sampling that is necessary to reflect connectivity. In addition, six Thalassinoides burrow sizes (expressed as the width of the Thalassinoides shafts), from 2.6 cm to 9 cm, were modelled by varying the cell size of the 1 m3D volume. In an example, the lower limit of the size range, 2.6 cm, may represent the Thalassinoides shafts of Jubaila in Saudi Arabia, whereas the upper limit of the size range, 9 cm, may represent the Thalassinoides shafts in the Miocene of southeast Spain. Decreasing the cell size in the 1 m3D volume results in an increase in the number of cells, as the smaller the cell size in the 1 m3D volume, the smaller the size of the Thalassinoides. This yields 1 m3D volumes with 1003, 2003, 3003, 4003, 5003, 6003 cells. In order to examine the results, six images from each of the six sides of the MPS models were extracted, for a total of 108 2D images with 1 marea. Further, each image represents one face. JMicroVision™ open-source software was used for image analysis to measure the width of the Thalassinoides shafts in each of the 108 images and use it as a proxy for Thalassinoides BS. Descriptive statistics of these measurements were calculated. Statistical parameters such as the mean (μ), median, minima, maxima, and standard deviation (σ) were used to represent the width of the Thalassinoides shaft that may be measured in each MPS model. The μ and σ were used to indicate the size of the Thalassinoides in each MPS model.

3 FIG.A 4 FIG.A 4 4 FIGS.A-C 4 4 FIGS.B andC 2 2 In order to simulate measurements that may be taken from core or image logs, vertical 1 m long columns in four corners and centre of the 18 MPS model cubes were selected as fixed locations for sampling. In particular, five sampling locations were selected, as shown inand. A systematic sampling of a randomized modelled Thalassinoides network was performed. At each one of these five sampling locations, six samples were extracted by cropping the model to progressively smaller column cross sections, as shown in. An area of a top of the column is the sample cross section (SCS), and the SCS ranges from 25 cmto 900 cm, by progressively increasing the side length of the top of the column in 5 cm steps, as specified in. Thus, in each 3D model cube (with a single BP and BS), there are 30 columnar samples. This yields a total of 540 digital samples to evaluate the range of BP, BS and SCS. The examination of the results helps for determining the cross-sectional area that is needed for a 1-meter-long sample to fully represent the burrow connectivity (proxy for permeability) from top to bottom of the sample. This is determined for a range of Thalassinoides BP and BS.

4 FIG.C 4 FIG.C To evaluate sample-scale-dependence of being able to measure connectivity of burrows, separate burrow connectivity analyses was performed on the 540 columnar digital samples as well as on the MPS model cubes. Burrow connectivity was analysed in Petrel™ 2020 using a connected volume function which defines cells that share adjacent faces. The volume of the largest connected burrow volume (LCBV) is used as a proxy for burrow connectivity. In an example, if the LCBV touches the upper and lower faces of the digital cube or columnar sample, then the model has vertical connected burrow volumes and high permeability. This analysis was first performed on the model cubes to verify that the burrows connected across the top and bottom of each model cube. Furthermore, the columnar samples were taken from each cube. For each columnar sample, the burrow connectivity determination was made separately. In order to best simulate measurements on actual samples, the vertical faces of the column were impermeable, and connectivity was only determined from the top to the bottom. Each connected burrow volume was given a unique colour code and code ranked by size, as shown in. In case the LCBV connected the upper face to the lower face, then the sample measures as permeable and may percolate fluid from top to bottom, as shown in. Further, in case the LCBV does not touch the upper and lower faces of the sample, then the burrow volumes may not provide permeability pathways from bottom to top of a measured sample, even if the larger sample is known to show such connectivity. Furthermore, two useful parameters for the LCBV are also documented: the length and the position. The length of the LCBV represents its vertical extent, whereas the position of the LCBV represents its location with respect to the top and bottom of the digital sample. For LCBV position, there are four possibilities as follows: touching only the top of the sample; touching only the bottom of the sample; touching both the top and bottom, and not touching the top or bottom.

3 FIG.C As stated above, in each of the 18 3DMPS models, with BP and BS, 5 digital samples were extracted with the same SCS and were used to determine if LCBV is connected across the upper and lower faces. If the LCBV showed connectivity, as explained in Example 3, the sample was assigned a score of 1. If not, the sample was assigned a score of 0. The probability of a digital sample with a particular BP and BS showing connectivity was calculated as the percentage of the samples with LCBV that showed connectivity from these five samples. A simple example is the five samples which were extracted from the MPS model of 1003 cells (9 cm burrows) and 25% BP. If the connectivity analysis showed that the LCBV in all five samples did not connect through the sample, then the probability of connectivity would be 0%. If all showed connectivity, then the probability of connectivity would be 100%. If three showed connectivity, then the probability of connectivity would be 60%. Probability of connectivity was also expressed by the exceedance probability (EP) of the LCBV length. The lengths of the LCBV were ranked from smallest to largest based on BP, BS and SCS, EP corresponding to each sample was calculated. The results provided three cross plots representing the EP of the ranked length of the LCBV against their actual length. The probability of the samples was calculated from the LCBV length data, specifying whether samples of a particular length may sample the LCBV, given specified SCS, BP, and BS, as provided in.

3 FIG.B In accordance with the present disclosure, binary logistic regression modelling was conducted in order to understand how BP, BS and SCS impact Thalassinoides connectivity of the digital samples, as shown in. Two categories of connectivity of the LCBV, connected burrow network (given the binary code of 1) and isolated burrow volume (given the binary code 0), were used as a binary dependent variable, whereas the BP, BS and SCS were used as independent variables. The binary logistic regression was performed using XLSTAT software. Results of the binary logistic regression include significance level (P-value) which is used to determine if there is a statistically significant association between the dependent (binary class of Thalassinoides connectivity) and the independent variables (BP, BS and SCS). The results further include beta coefficients (β) which are used to calculate the factor score for the independent variables (BP, BS and SCS) for the probability of Thalassinoides connectivity, and probability equation from logistic regression results, which may be used to calculate the probability of LCBV, in a 1-meter-long sample, connecting across the top and bottom of the sample, for a given BP, BS and SCS.

The dataset, which includes results from 540 samples, was randomly divided into two subsets. A total of 432 samples (about 80% of the samples) were used as a training set for the regression modelling, whereas 108 samples (about 20% of the samples) were used as a training set for external validation, as shown in Table 1A. The training dataset was used to run the logistic regression analysis, and validation dataset was used to validate the results of the logistic regression.

Validation was conducted by comparing the predicted binary class of the Thalassinoides connectivity versus isolated burrow volume from the logistic regression equation to their actual class. Such comparison helped to evaluate the sensitivity and specificity of the logistic regression results and allowed for assessment if the method can be used as a tool for predicting the Thalassinoides connectivity. For such evaluation, receiver operating characteristics (ROC) method was used, which plots the specificity against 1-specificity of observed versus predicted values. Two approaches for validation were conducted (internal and external). In the internal validation, the training dataset (432 samples) was used for validation, whereas in the external validation, the validation dataset (108 samples) was used. The details of sample sets for training validation, as well other experimental data pertaining to validation is provided in the Tables 1A-1F. It is noteworthy that the provided data is non-limiting in nature.

TABLE 1A Data sampling (training and validation set - 540 samples) BS BP SCS BC BS BP SCS BC 9 75 10 1 9 75 100 1 2.6 50 20 1 2.6 50 400 1 3.4 75 25 1 3.4 75 625 1 2.6 25 5 0 2.6 25 25 0 2.6 25 15 1 2.6 25 225 1 4 25 10 0 4 25 100 0 2.7 25 30 1 2.7 25 900 1 9 75 15 1 9 75 225 1 9 25 15 0 9 25 225 0 9 75 20 1 9 75 400 1 5.1 25 30 1 5.1 25 900 1 2.6 50 20 1 2.6 50 400 1 2.6 75 15 1 2.6 75 225 1 2.7 75 30 1 2.7 75 900 1 2.7 25 15 1 2.7 25 225 1 2.6 25 20 1 2.6 25 400 1 5.1 50 30 1 5.1 50 900 1 4 25 20 1 4 25 400 1 5.1 50 5 0 5.1 50 25 0 2.7 25 25 1 2.7 25 625 1 2.6 75 25 1 2.6 75 625 1 4 50 20 1 4 50 400 1 2.7 25 20 1 2.7 25 400 1 9 25 20 0 9 25 400 0 2.7 50 10 0 2.7 50 100 0 2.6 75 5 1 2.6 75 25 1 3.4 75 15 1 3.4 75 225 1 9 50 10 0 9 50 100 0 9 50 10 0 9 50 100 0 9 50 25 1 9 50 625 1 2.6 25 30 1 2.6 25 900 1 2.6 50 10 1 2.6 50 100 1 9 50 20 1 9 50 400 1 9 50 30 1 9 50 900 1 5.1 25 20 1 5.1 25 400 1 9 50 5 0 9 50 25 0 5.1 25 30 1 5.1 25 900 1 3.4 75 25 1 3.4 75 625 1 5.1 50 20 1 5.1 50 400 1 5.1 50 15 0 5.1 50 225 0 3.4 75 20 1 3.4 75 400 1 5.1 25 10 0 5.1 25 100 0 3.4 50 5 0 3.4 50 25 0 5.1 75 10 1 5.1 75 100 1 5.1 50 15 0 5.1 50 225 0 4 50 5 0 4 50 25 0 9 75 15 1 9 75 225 1 9 25 20 0 9 25 400 0 4 75 10 1 4 75 100 1 3.4 75 25 1 3.4 75 625 1 4 75 10 1 4 75 100 1 2.6 75 30 1 2.6 75 900 1 5.1 75 30 1 5.1 75 900 1 4 50 15 1 4 50 225 1 2.6 50 30 1 2.6 50 900 1 9 50 5 0 9 50 25 0 3.4 50 20 1 3.4 50 400 1 9 75 15 1 9 75 225 1 3.4 50 30 1 3.4 50 900 1 3.4 50 5 0 3.4 50 25 0 4 75 15 1 4 75 225 1 5.1 75 20 1 5.1 75 400 1 9 75 30 1 0 75 900 1 2.6 25 10 0 2.6 25 100 0 2.7 50 15 1 2.7 50 225 1 3.4 50 5 0 3.4 50 25 0 2.6 25 10 0 2.6 25 100 0 5.1 75 5 0 5.1 75 25 0 2.6 25 5 0 2.6 25 25 0 2.6 25 0 0 2.6 25 100 0 2.6 50 25 1 2.6 50 625 1 2.6 25 30 1 2.6 25 900 1 9 75 5 1 9 75 25 1 2.7 75 25 1 2.7 75 625 1 2.7 75 25 1 2.7 75 625 1 2.7 50 25 1 2.7 50 625 1 2.6 25 25 1 2.6 25 625 1 3.4 75 10 1 3.4 75 100 1 3.4 25 30 1 3.4 25 900 1 2.6 75 30 1 2.6 75 900 1 4 75 25 1 4 75 625 1 2.7 75 30 1 2.7 75 900 1 2.6 50 20 1 2.6 50 400 1 9 25 20 0 9 25 400 0 9 25 5 0 9 25 25 0 5.1 75 20 1 5.1 75 400 1 2.7 50 5 0 2.7 50 25 0 3.4 50 10 1 3.4 50 100 1 2.6 50 5 0 2.6 50 25 0 5.1 25 5 0 5.1 25 25 0 4 75 15 1 4 75 225 1 5.1 25 5 0 5.1 25 25 0 2.7 50 20 1 2.7 50 400 1 2.7 75 5 1 2.7 75 25 1 2.6 25 25 1 2.6 25 625 1 3.4 25 20 1 3.4 25 400 1 2.6 50 10 1 2.6 50 100 1 9 75 30 1 9 75 900 1 2.6 25 20 1 2.6 25 400 1 2.7 25 5 0 2.7 25 25 0 3.4 50 30 1 3.4 50 900 1 9 75 20 1 0 75 400 1 5.1 50 20 1 5.1 50 400 1 9 50 10 0 9 50 100 0 2.6 75 25 1 2.6 75 625 1 5.1 75 20 1 5.1 75 400 1 3.4 50 15 1 3.4 50 225 1 4 25 15 1 4 25 225 1 3.4 50 15 1 3.4 50 225 1 2.6 50 15 1 2.6 50 225 1 4 25 30 1 4 25 900 1 4 50 15 1 4 50 225 1 3.4 75 10 1 3.4 75 100 1 9 50 15 0 9 50 225 0 4 25 10 1 4 25 100 1 3.4 25 5 0 3.4 25 25 0 5.1 75 30 1 5.1 75 900 1 2.6 50 25 1 2.6 50 625 1 5.1 50 30 1 5.1 50 900 1 5.1 25 25 1 5.1 25 625 1 9 75 25 1 9 75 625 1 2.6 50 5 0 2.6 50 25 0 3.4 25 15 0 3.4 25 225 0 9 25 5 0 9 25 25 0 2.6 25 15 1 2.6 25 225 1 9 25 20 0 9 25 400 0 2.7 25 10 0 2.7 25 100 0 5.1 50 10 0 5.1 50 100 0 2.6 50 15 1 2.6 50 225 1 2.7 25 15 1 2.7 25 225 1 2.7 25 5 0 2.7 25 25 0 9 75 15 1 9 75 225 1 2.6 75 5 1 2.6 75 25 1 5.1 25 5 0 5.1 25 25 0 2.7 75 10 1 2.7 75 100 1 5.1 25 30 1 5.1 25 900 1 9 25 10 0 9 25 100 0 2.6 75 30 1 2.6 75 900 1 2.7 75 30 1 2.7 75 900 1 9 25 30 1 9 25 900 1 3.4 25 5 0 3.4 25 25 0 9 50 15 1 9 50 225 1 2.7 50 25 1 2.7 50 625 1 5.1 25 5 0 5.1 25 25 0 2.6 75 15 1 2.6 75 225 1 2.6 75 10 1 2.6 75 100 1 5.1 25 25 1 5.1 25 625 1 3.4 75 20 1 3.4 75 400 1 4 25 5 0 4 25 25 0 2.6 75 25 1 2.6 75 625 1 2.7 25 25 1 2.7 25 625 1 2.6 75 30 1 2.6 75 900 1 9 50 25 0 9 50 625 0 2.7 25 25 1 2.7 25 625 1 2.7 75 20 1 2.7 75 400 1 2.7 50 25 1 2.7 50 625 1 5.1 75 25 1 5.1 75 625 1 3.4 50 25 1 3.4 50 625 1 2.6 50 30 1 2.6 50 900 1 9 75 5 0 9 75 25 0 5.1 75 25 1 5.1 75 625 1 9 25 15 0 9 25 225 0 2.7 75 15 1 2.7 75 225 1 4 25 30 1 4 25 900 1 4 75 5 1 4 75 25 1 9 25 25 0 9 25 625 0 5.1 50 10 0 5.1 50 100 0 2.7 75 5 1 2.7 75 25 1 9 25 5 0 9 25 25 0 3.4 25 25 1 3.4 25 625 1 9 25 25 0 9 25 625 0 3.4 75 30 1 3.4 75 900 1 5.1 25 20 0 5.1 25 400 0 3.4 50 20 1 3.4 50 400 1 9 75 30 1 9 75 900 1 2.7 75 25 1 2.7 75 625 1 4 75 25 1 4 75 625 1 2.6 75 5 1 2.6 75 25 1 9 50 30 1 9 50 900 1 9 25 15 0 9 25 225 0 5.1 75 5 1 5.1 75 25 1 5.1 75 20 1 5.1 75 400 1 4 50 20 1 4 50 400 1 2.7 75 10 1 2.7 75 100 1 5.1 50 30 1 5.1 50 900 1 3.4 50 25 1 3.4 50 625 1 2.7 50 20 1 2.7 50 400 1 5.1 25 30 1 5.1 25 900 1 2.7 50 15 1 2.7 50 225 1 4 75 10 1 4 75 100 1 2.6 25 10 1 2.6 25 100 1 9 25 30 0 9 25 900 0 5.1 50 10 0 5.1 50 100 0 9 50 5 0 9 50 25 0 2.7 75 30 1 2.7 75 900 1 5.1 25 5 0 5.1 25 25 0 4 25 25 1 4 25 625 1 5.1 75 10 1 5.1 75 100 1 3.4 75 15 1 3.4 75 225 1 2.7 50 30 1 2.7 50 900 1 4 25 10 0 4 25 100 0 2.7 75 15 1 2.7 75 225 1 2.6 50 5 0 2.6 50 25 0 2.6 75 20 1 2.6 75 400 1 2.7 25 20 1 2.7 25 400 1 9 50 25 1 9 50 625 1 4 75 25 1 4 75 625 1 9 50 5 0 9 50 25 0 2.7 50 20 1 2.7 50 400 1 2.7 25 5 0 2.7 25 25 0 2.7 50 5 0 2.7 50 25 0 3.4 75 20 1 3.4 75 400 1 2.7 75 0 1 2.7 75 100 1 9 50 25 1 9 50 625 1 9 75 10 0 9 75 100 0 5.1 50 20 1 5.1 50 400 1 5.1 50 15 0 5.1 50 225 0 2.7 25 20 1 2.7 25 400 1 4 25 15 0 4 25 225 0 2.7 50 10 1 2.7 50 100 1 5.1 25 15 0 5.1 25 225 0 9 75 5 0 9 75 25 0 2.7 25 20 1 2.7 25 400 1 3.4 75 20 1 3.4 75 400 1 5.1 75 30 1 5.1 75 900 1 2.6 50 15 1 2.6 50 225 1 4 25 10 0 4 25 100 0 4 50 5 0 4 50 25 0 3.4 50 15 1 3.4 50 225 1 2.6 25 25 1 2.6 25 625 1 2.6 50 5 0 2.6 50 25 0 2.7 50 5 0 2.7 50 25 0 9 75 5 0 9 75 25 0 2.7 25 15 1 2.7 25 225 1 4 25 30 1 4 25 900 1 3.4 25 10 0 3.4 25 100 0 5.1 25 10 0 5.1 25 100 0 2.7 75 20 1 2.7 75 400 1 2.6 25 5 0 2.6 25 25 0 2.7 50 5 0 2.7 50 25 0 4 25 10 1 4 25 100 1 3.4 25 25 1 3.4 25 625 1 5.1 75 15 1 5.1 75 225 1 2.6 25 15 1 2.6 25 225 1 2.6 50 25 1 2.6 50 625 1 3.4 75 30 1 3.4 75 900 1 5.1 75 30 1 5.1 75 900 1 2.6 75 10 1 2.6 75 100 1 4 75 5 0 4 75 25 0 2.6 25 20 1 2.6 25 400 1 3.4 50 5 0 3.4 50 25 0 9 25 25 0 9 25 625 0 5.1 25 20 0 5.1 25 400 0 3.4 50 20 1 3.4 50 400 1 2.7 25 15 0 2.7 25 225 0 5.1 25 30 0 5.1 25 900 0 5.1 50 25 1 5.1 50 625 1 3.4 25 5 0 3.4 25 25 0 4 50 15 1 4 50 225 1 5.1 75 25 1 5.1 75 625 1 9 25 15 0 9 25 225 0 3.4 50 25 1 3.4 50 625 1 5.1 25 20 0 5.1 25 400 0 3.4 75 25 1 3.4 75 625 1 5.1 25 15 0 5.1 25 225 0 5.1 25 15 0 5.1 25 225 0 9 50 15 0 9 50 225 0 2.6 50 10 1 2.6 50 100 1 2.7 75 5 0 2.7 75 25 0 2.7 50 10 0 2.7 50 100 0 9 50 10 0 9 50 100 0 5.1 50 25 1 5.1 50 625 1 2.7 75 25 1 2.7 75 625 1 2.6 25 20 1 2.6 25 400 1 5.1 50 5 0 5.1 50 25 0 3.4 50 10 1 3.4 50 100 1 3.4 75 25 1 3.4 75 625 1 5.1 75 5 0 5.1 75 25 0 2.7 25 30 1 2.7 25 900 1 9 50 30 1 9 50 900 1 2.7 25 5 0 2.7 25 25 0 9 50 10 0 9 50 100 0 5.1 50 5 0 5.1 50 25 0 3.4 25 20 0 3.4 25 400 0 5.1 50 25 1 5.1 50 625 1 2.6 50 25 1 2.6 50 625 1 3.4 25 5 0 3.4 25 25 0 5.1 75 25 1 5.1 75 625 1 2.6 25 25 1 2.6 25 625 1 9 75 10 1 9 75 100 1 2.6 50 25 1 2.6 50 625 1 3.4 25 20 1 3.4 25 400 1 5.1 75 15 1 5.1 75 225 1 3.4 25 10 0 3.4 25 100 0 4 50 30 1 4 50 900 1 4 75 20 1 4 75 400 1 3.4 75 15 1 3.4 75 225 1 2.7 25 25 1 2.7 25 625 1 4 50 25 1 4 50 625 1 5.1 75 30 1 5.1 75 900 1 4 50 10 0 4 50 100 0 2.7 25 10 0 2.7 25 100 0 5.1 50 10 0 5.1 50 100 0 4 25 20 0 4 25 400 0 5.1 75 10 0 5.1 75 100 0 2.6 75 20 1 2.6 75 400 1 2.6 75 15 1 2.6 75 225 1 5.1 50 30 1 5.1 50 900 1 2.7 75 30 1 2.7 75 900 1 2.7 75 20 1 2.7 75 400 1 4 50 10 0 4 50 100 0 9 50 15 0 9 50 225 0 2.7 75 10 1 2.7 75 100 1 2.7 50 30 1 2.7 50 900 1 4 25 10 0 4 25 100 0 3.4 75 5 0 3.4 75 25 0 3.4 25 25 1 3.4 25 625 1 2.7 50 30 1 2.7 50 900 1 5.1 25 20 0 5.1 25 400 0 2.6 25 5 0 2.6 25 25 0 4 25 20 1 4 25 400 1 3.4 75 10 1 3.4 75 100 1 3.4 25 30 1 3.4 25 900 1 2.7 50 30 1 2.7 50 900 1 2.7 50 25 1 2.7 50 625 1 5.1 25 25 0 5.1 25 625 0 2.6 50 5 0 2.6 50 25 0 4 75 30 1 4 75 900 1 2.6 50 10 1 2.6 50 100 1 4 50 25 1 4 50 625 1 2.7 75 10 1 2.7 75 100 1 2.6 75 15 1 2.6 75 225 1 5.1 75 15 1 5.1 75 225 1 5.1 25 25 0 5.1 25 625 0 2.6 75 15 1 2.6 75 225 1 9 50 20 1 9 50 400 1 5.1 25 10 0 5.1 25 100 0 9 50 15 0 9 50 225 0 2.7 25 20 1 2.7 25 400 1 9 25 25 0 9 25 625 0 3.4 75 5 1 3.4 75 25 1 4 25 15 0 4 25 225 0 3.4 25 25 1 3.4 25 625 1 2.7 75 20 1 2.7 75 400 1 2.6 25 30 1 2.6 25 900 1 2.7 50 15 1 2.7 50 225 1 9 25 25 0 9 25 625 0 3.4 25 15 0 3.4 25 225 0 3.4 75 5 1 3.4 75 25 1 9 25 30 0 9 25 900 0 5.1 75 25 1 5.1 75 625 1 5.1 50 15 1 5.1 50 225 1 3.4 50 15 1 3.4 50 225 1 4 50 10 0 4 50 100 0 5.1 25 15 1 5.1 25 225 1 2.7 50 20 1 2.7 50 400 1 5.1 50 30 1 5.1 50 900 1 3.4 50 15 1 3.4 50 225 1 3.4 75 5 0 3.4 75 25 0 2.6 25 30 1 2.6 25 900 1 4 75 20 1 4 75 400 1 4 75 20 1 4 75 400 1 4 50 30 1 4 50 900 1 2.6 50 20 1 2.6 50 400 1 3.4 50 25 1 3.4 50 625 1 2.7 25 5 0 2.7 25 25 0 3.4 75 20 1 3.4 75 400 1 2.6 50 15 1 2.6 50 225 1 9 75 25 1 9 75 625 1 4 75 30 1 4 75 900 1 9 25 15 0 9 25 225 0 3.4 50 25 1 3.4 50 625 1 2.7 50 5 0 2.7 50 25 0 3.4 25 10 0 3.4 25 100 0 4 50 15 1 4 50 225 1 9 50 5 0 9 50 25 0 2.7 75 5 1 2.7 75 25 1 5.1 75 10 1 5.1 75 100 1 3.4 25 15 1 3.4 25 225 1 3.4 25 30 1 3.4 25 900 1 4 50 20 1 4 50 400 1 2.6 75 5 1 2.6 75 25 1 4 50 5 0 4 50 25 0 9 25 30 1 9 25 900 1 4 75 10 1 4 75 100 1 2.6 75 20 1 2.6 75 400 1 3.4 25 30 1 3.4 25 900 1 4 75 25 1 4 75 625 1 3.4 75 30 1 3.4 75 900 1 3.4 75 15 1 3.4 75 225 1 2.6 25 15 0 2.6 25 225 0 2.6 75 25 1 2.6 75 625 1 4 75 10 1 4 75 100 1 5.1 50 15 1 5.1 50 225 1 4 25 25 1 4 25 625 1 4 25 5 0 4 25 25 0 2.6 25 10 1 2.6 25 100 1 4 50 25 1 4 50 625 1 5.1 50 20 1 5.1 50 400 1 3.4 25 10 0 3.4 25 100 0 4 25 25 1 4 25 625 1 5.1 75 10 0 5.1 75 100 0 2.6 25 15 1 2.6 25 225 1 3.4 50 10 1 3.4 50 100 1 3.4 75 10 1 3.4 75 100 1 2.6 75 20 1 2.6 75 400 1 4 75 25 1 4 75 625 1 9 75 5 1 9 75 25 1 3.4 75 15 1 3.4 75 225 1 2.7 50 20 1 2.7 50 400 1 3.4 75 30 1 3.4 75 900 1 4 50 25 1 4 50 625 1 3.4 50 30 1 3.4 50 900 1 4 75 30 1 4 75 900 1 4 75 30 1 4 75 900 1 4 75 5 0 4 75 25 0 9 75 15 1 9 75 225 1 2.6 25 25 1 2.6 25 625 1 9 25 10 0 9 25 100 0 4 75 5 0 4 75 25 0 2.6 25 30 1 2.6 25 900 1 9 50 20 0 9 50 400 0 2.6 50 15 1 2.6 50 225 1 5.1 75 15 1 5.1 75 225 1 9 75 30 1 9 75 900 1 3.4 25 15 1 3.4 25 225 1 3.4 25 10 0 3.4 25 100 0 5.1 50 5 0 5.1 50 25 0 2.7 75 15 1 2.7 75 225 1 2.6 75 30 1 2.6 75 900 1 2.6 75 10 1 2.6 75 100 1 2.7 75 20 1 2.7 75 400 1 5.1 25 15 0 5.1 25 225 0 4 50 5 0 4 50 25 0 4 75 15 1 4 75 225 1 2.6 75 25 1 2.6 75 625 1 4 75 15 1 4 75 225 1 2.6 75 5 1 2.6 75 25 1 3.4 25 15 0 3.4 25 225 0 9 25 30 0 9 25 900 0 9 25 20 0 9 25 400 0 4 50 20 1 4 50 400 1 4 75 20 1 4 75 400 1 3.4 25 20 1 3.4 25 400 1 2.6 50 10 1 2.6 50 100 1 2.7 50 10 0 2.7 50 100 0 4 75 20 1 4 75 400 1 2.6 25 20 1 2.6 25 400 1 2.7 50 15 1 2.7 50 225 1 5.1 25 10 0 5.1 25 100 0 9 75 20 1 9 75 400 1 9 25 10 0 9 25 100 0 5.1 75 5 0 5.1 75 25 0 9 50 30 1 9 50 900 1 4 25 5 0 4 25 25 0 3.4 75 5 1 3.4 75 25 1 2.7 25 30 1 2.7 25 900 1 4 75 15 1 4 75 225 1 2.7 50 30 1 2.7 50 900 1 9 75 30 1 9 75 900 1 4 25 30 1 4 25 900 1 4 25 5 0 4 25 25 0 2.7 75 15 1 2.7 75 225 1 2.7 25 30 1 2.7 25 900 1 9 75 10 1 9 75 100 1 4 50 5 0 4 50 25 0 9 75 25 1 9 75 625 1 2.6 25 5 0 2.6 25 25 0 2.6 50 20 1 2.6 50 400 1 2.6 75 10 1 2.6 75 100 1 4 25 15 0 4 25 225 0 4 50 20 1 4 50 400 1 3.4 25 25 1 3.4 25 625 1 9 75 10 1 9 75 100 1 3.4 50 5 0 3.4 50 25 0 5.1 75 15 1 5.1 75 225 1 5.1 50 20 1 5.1 50 400 1 2.6 75 20 1 2.6 75 400 1 9 75 20 1 9 75 400 1 3.4 50 10 1 3.4 50 100 1 5.1 25 25 1 5.1 25 625 1 4 75 30 1 4 75 900 1 4 25 30 1 4 25 900 1 9 25 10 0 9 25 100 0 4 50 25 1 4 50 625 1 2.7 75 15 1 2.7 75 225 1 3.4 25 20 1 3.4 25 400 1 3.4 50 20 1 3.4 50 400 1 4 25 25 0 4 25 625 0 2.7 25 30 1 2.7 25 900 1 3.4 50 20 1 3.4 50 400 1 3.4 75 10 1 3.4 75 100 1 2.7 50 25 1 2.7 50 625 1 9 25 5 0 9 25 25 0 4 50 30 1 4 50 900 1 9 25 5 0 9 25 25 0 2.7 25 15 1 2.7 25 225 1 5.1 50 5 0 5.1 50 25 0 4 25 15 0 4 25 225 0 9 50 20 0 9 50 400 0 5.1 50 25 1 5.1 50 625 1 4 25 25 1 4 25 625 1 5.1 75 20 1 5.1 75 400 1 4 25 20 1 4 25 400 1 9 75 25 1 9 75 625 1 3.4 50 30 1 3.4 50 900 1 2.7 25 25 1 2.7 25 625 1 9 50 20 0 9 50 400 0 9 25 10 0 9 25 100 0 9 50 30 1 9 50 900 1 9 50 25 1 9 50 625 1 3.4 50 30 1 3.4 50 900 1 2.6 50 30 1 2.6 50 900 1 4 25 20 1 4 25 400 1 2.7 25 10 0 2.7 25 100 0 3.4 25 5 0 3.4 25 25 0 5.1 75 5 0 5.1 75 25 0 4 25 5 0 4 25 25 0 4 75 5 0 4 75 25 0 2.6 50 30 1 2.6 50 900 1 2.7 25 10 0 2.7 25 100 0 4 50 10 1 4 50 100 1 3.4 50 10 1 3.4 50 100 1 2.6 50 30 1 2.6 50 900 1 3.4 25 30 1 3.4 25 900 1 2.7 75 5 0 2.7 75 25 0 2.7 25 10 0 2.7 25 100 0 4 50 30 1 4 50 900 1 5.1 50 25 1 5.1 50 625 1 4 50 15 1 4 50 225 1 4 50 10 0 4 50 100 0 5.1 50 10 0 5.1 50 100 0 2.7 50 15 1 2.7 50 225 1 4 50 30 1 4 50 900 1 2.7 75 25 1 2.7 75 625 1 9 75 25 1 9 75 625 1 9 75 20 1 9 75 400 1 2.6 75 10 1 2.6 75 100 1 3.4 75 30 1 3.4 75 900 1 2.7 50 10 1 2.7 50 100 1

TABLE 1B Summary statistics (quantitative data) Obs. Obs. with without missing missing Std. Variable Observations data data Minimum Maximum Mean deviation BS 432 0 432 2.6 9 4.422 2.177 BP 432 0 432 25 75 50.289 20.54 SCS 432 0 432 25 900 376.157 303.477

TABLE 1C Summary statistics (quantitative data/validation) Obs. Obs. with without missing missing Std. Variable Observations data data Minimum Maximum Mean deviation BS 108 0 108 2.6 9 4.633 2.279 BP 108 0 108 25 75 48.843 20.042 SCS 108 0 108 25 900 391.204 315.049

TABLE 1D Summary statistics (qualitative data) Variable Categories Counts Frequencies % BC 0 139 139 32.176 1 293 293 67.824

TABLE 1E Summary statistics (Qualitative data/validation) Variable Categories Counts Frequencies % BC 0 34 34 31.481 1 74 74 68.519

TABLE 1F Correlation matrix BS BP SCS BC BS 1 −0.019 −0.008 −0.303 BP −0.019 1 0.012 0.384 SCS −0.008 0.012 1 0.496 BC −0.303 0.384 0.496 1

5 5 FIGS.A-F 6 6 FIGS.A-F 6 6 FIGS.A-F 3 The MPS modelling produces variation in BS by varying cell size of the models (consequently increasing cell number), as shown in,and Table 2. Results of image analysis using JMicroVision™ help to evaluate BS of the modelled Thalassinoides. For example, the measured diameter of Thalassinoides (BS) on the images of the model with 1003 cells yields a mean of 9.0±4.6 cm; analysis of images from the 6003-cell model yields a mean BS of 2.6±1.3 cm, as listed in Table 2. Histograms of the measured diameter of the modelled Thalassinoides showed progressively smaller ranges, while increasing the number of the cells in the MPS models, as shown in. This result is in line with the observation that the Thalassinoides network is better represented in the 1 mmodels, when cell size is large, and BS is small. Table 2 shows the mean of the measured diameter of Thalassinoides in each MPS model, which is used as a proxy for BS in the following sections.

TABLE 2 Thalassinoides Mean of the cross section of network in each MPS model Cell size of Thalassinoides Mean of the the MPS model diameter (cm) 100 9 200 5.1 300 4 400 3.4 500 2.7 600 2.6

3 3 5 5 FIGS.A-F Results of connectivity analyses that were run on the full 1 mvolumes of the 18 MPS models indicate that the Thalassinoides in these models connects across the top and bottom of the 1 m3D models, as shown in. In each model, more than 95% of the total volume of Thalassinoides was connected as part of a LCBV. When the same analysis was run on the 540 columnar samples, taken from these 18 MPS models, however, 173 of the samples (about 32%) showed no top-to-bottom connectivity for the Thalassinoides. This clearly indicates that there is scale dependence in how Thalassinoides connectivity must be sampled.

7 7 FIGS.A-F 7 7 FIGS.A-F 8 8 FIGS.A-C The BS, BP, and SCS controls on probability of Thalassinoides connectivity in the 540 samples may be illustrated graphically, as shown in. The results show that for a given BS and SCS, increasing BP results in increased probability of Thalassinoides samples showing connectivity. In contrast, the results show that for a given BP and SCS, increased BS decreases the probability of samples showing the Thalassinoides connectivity, as listed in Table 3A and 3B. Also, for a given BP and BS, increasing the SCS results in an increased probability of samples showing Thalassinoides connectivity, as shown in, Table 3A and Table 3B. Similar results may be shown in cross plots of the lengths of the LCBV, as shown in Table 3A and 3B, Tables 4A-4D, and Tables 5A-5D against their EP depicted in. In these plots, the curves flatten out at 100 cm where the LCBV of Thalassinoides has reached the maximum length, showing vertical connectivity in the sample. It may be noted in these plots that the point at which the LCBV of Thalassinoides reaches the 100 cm flattening varies depending on the BP, BS and SCS. These graphical analyses of the results visually highlight the importance of BP, BS, and sample size as variables to allow Thalassinoides connectivity to be sampled and properly represented.

9 9 FIGS.A-C 9 9 FIGS.A-C 10 FIG. The results of binary logistic regression confirm and quantify the graphic analysis results about Thalassinoides connectivity, and the relationships among BP, BS and SCS, as shown in. The binary logistic regression results quantify the impact of each one of these variables for 1-meter-long samples. Results of the binary logistic regression, as listed in Tables 3A-3I, indicate that BP, BS and SCS are significant predictors for representing Thalassinoides connectivity in samples (P-value for all is <0.05). The B coefficients indicate that there is a negative impact of BS on Thalassinoides connectivity in samples, whereas there is a positive impact of BP and SCS on Thalassinoides connectivity in samples. Standardized β coefficients indicate that SCS has the highest impact on representation of Thalassinoides connectivity in the digital samples, followed by BP, and then BS, as shown in. An important product of the logistic regression modelling is a probability equation 1 for samples to represent Thalassinoides connectivity in a 1-meter-long sample, as shown in. The equation is as follows:

9 FIG.B 9 FIG.C The aforementioned probability equation for the Thalassinoides connectivity can be used to assess the volume/SCS of 1-meter-long sample required to represent the connectivity of Thalassinoides, provided that the BP and BS are known. The internal validation of the results of the logistic regression modelling using the ROC curve for the predictive probability of the model with the training data revealed an area under the curve of 0.958 with P less than 0.05, as shown in. When using the probability equation to predict Thalassinoides connectivity for the 108 samples of the external validation data set, the results revealed 85.29% specificity and 94.59 sensitivity. The accuracy of the probability model is about 91.67%. The ROC of the external validation using the 108 samples is 0.969, as shown in.

TABLE 3A Predictions and residuals (variable BC) Observation Obs1 1 0 0.756 0.244 Obs2 1 1 0.028 0.972 Obs3 1 1 0.001 0.999 Obs4 0 0 0.914 0.086 Obs5 1 0 0.606 0.394 Obs6 0 0 0.926 0.074 Obs7 1 1 0.002 0.998 Obs8 1 1 0.481 0.519 Obs9 0 0 0.989 0.011 Obs10 1 1 0.146 0.854 Obs11 1 1 0.011 0.989 Obs12 1 1 0.028 0.972 Obs13 1 1 0.016 0.984 Obs14 1 1 0 1 Obs15 1 0 0.621 0.379 Obs16 1 1 0.221 0.779 Obs17 1 1 0.001 0.999 Obs18 1 1 0.408 0.592 Obs19 0 0 0.841 0.159 Obs20 1 1 0.033 0.967 Obs21 1 1 0 1 Obs22 1 1 0.066 0.934 Obs23 1 1 0.232 0.768 Obs24 0 0 0.942 0.058 Obs25 0 1 0.36 0.64 Obs26 1 1 0.1 0.9 Obs27 1 1 0.026 0.974 Obs28 0 0 0.968 0.032 Obs29 0 0 0.968 0.032 Obs30 1 1 0.16 0.84 Obs31 1 1 0.002 0.998 Obs32 1 1 0.345 0.655 Obs33 1 0 0.626 0.374 Obs34 1 1 0.013 0.987 Obs35 1 0 0.58 0.42 Obs36 0 0 0.984 0.016 Obs37 1 1 0.011 0.989 Obs38 1 1 0.001 0.999 Obs39 1 1 0.124 0.876 Obs40 0 1 0.434 0.566 Obs41 1 1 0.005 0.995 Obs42 0 0 0.962 0.038 Obs43 0 0 0.643 0.357 Obs44 1 1 0.208 0.792 Obs45 0 1 0.434 0.566 Obs46 0 0 0.725 0.275 Obs47 1 1 0.481 0.519 Obs48 0 0 0.942 0.058 Obs49 1 1 0.116 0.884 Obs50 1 1 0.001 0.999 Obs51 1 1 0.116 0.884 Obs52 1 1 0 1 Obs53 1 1 0 1 Obs54 1 1 0.277 0.723 Obs55 1 1 0 1 Obs56 0 0 0.984 0.016 Obs57 1 1 0.046 0.954 Obs58 1 1 0.481 0.519 Obs59 1 1 0 1 Obs60 0 0 0.643 0.357 Obs61 1 1 0.038 0.962 Obs62 1 1 0.014 0.986 Obs63 1 1 0.001 0.999 Obs64 0 0 0.837 0.163 Obs65 1 1 0.144 0.856 Obs66 0 0 0.643 0.357 Obs67 0 0 0.837 0.163 Obs68 0 1 0.352 0.648 Obs69 0 0 0.914 0.086 Obs70 0 0 0.837 0.163 Obs71 1 1 0.003 0.997 Obs72 1 1 0.002 0.998 Obs73 1 0 0.865 0.135 Obs74 1 1 0 1 Obs75 1 1 0 1 Obs76 1 1 0.004 0.996 Obs77 1 1 0.031 0.969 Obs78 1 1 0.082 0.918 Obs79 1 1 0.004 0.996 Obs80 1 1 0 1 Obs81 1 1 0.001 0.999 Obs82 1 1 0 1 Obs83 1 1 0.028 0.972 Obs84 0 0 0.942 0.058 Obs85 0 0 0.998 0.002 Obs86 1 1 0.014 0.986 Obs87 0 0 0.537 0.463 Obs88 1 1 0.466 0.534 Obs89 0 0 0.521 0.479 Obs90 0 0 0.981 0.019 Obs91 1 1 0.038 0.962 Obs92 0 0 0.981 0.019 Obs93 1 1 0.03 0.97 Obs94 1 1 0.106 0.894 Obs95 1 1 0.031 0.969 Obs96 1 1 0.32 0.68 Obs97 1 1 0.345 0.655 Obs98 1 1 0.001 0.999 Obs99 1 1 0.221 0.779 Obs100 0 0 0.919 0.081 Obs101 1 1 0 1 Obs102 1 1 0.146 0.854 Obs103 1 1 0.124 0.876 Obs104 0 0 0.968 0.032 Obs105 1 1 0 1 Obs106 1 1 0.014 0.986 Obs107 1 1 0.207 0.793 Obs108 1 0 0.789 0.211 Obs109 1 1 0.207 0.793 Obs110 1 1 0.136 0.864 Obs111 1 1 0.005 0.995 Obs112 1 1 0.277 0.723 Obs113 1 1 0.082 0.918 Obs114 0 0 0.901 0.099 Obs115 1 0 0.926 0.074 Obs116 0 0 0.946 0.054 Obs117 1 1 0 1 Obs118 1 1 0.003 0.997 Obs119 1 1 0.001 0.999 Obs120 1 1 0.136 0.864 Obs121 1 1 0.019 0.981 Obs122 0 0 0.521 0.479 Obs123 0 0 0.719 0.281 Obs124 0 0 0.998 0.002 Obs125 1 0 0.606 0.394 Obs126 0 0 0.942 0.058 Obs127 0 0 0.846 0.154 Obs128 0 0 0.719 0.281 Obs129 1 1 0.136 0.864 Obs130 1 0 0.621 0.379 Obs131 0 0 0.919 0.081 Obs132 1 1 0.481 0.519 Obs133 1 1 0.1 0.9 Obs134 0 0 0.981 0.019 Obs135 1 1 0.054 0.946 Obs136 1 1 0.011 0.989 Obs137 0 0 0.997 0.003 Obs138 1 1 0 1 Obs139 1 1 0 1 Obs140 1 1 0.116 0.884 Obs141 0 0 0.946 0.054 Obs142 1 0 0.901 0.099 Obs143 1 1 0.004 0.996 Obs144 0 0 0.981 0.019 Obs145 1 1 0.016 0.984 Obs146 1 1 0.051 0.949 Obs147 1 1 0.136 0.864 Obs148 1 1 0.005 0.995 Obs149 0 0 0.963 0.037 Obs150 1 1 0 1 Obs151 1 1 0.033 0.967 Obs152 1 1 0 1 Obs153 0 1 0.16 0.84 Obs154 1 1 0.033 0.967 Obs155 1 1 0.003 0.997 Obs156 1 1 0.004 0.996 Obs157 1 1 0.002 0.998 Obs158 1 1 0.005 0.995 Obs159 1 1 0 1 Obs160 0 0 0.865 0.135 Obs161 1 1 0.002 0.998 Obs162 0 0 0.989 0.011 Obs163 1 1 0.017 0.983 Obs164 1 1 0.005 0.995 Obs165 1 1 0.213 0.787 Obs166 0 0 0.65 0.35 Obs167 0 0 0.719 0.281 Obs168 1 1 0.106 0.894 Obs169 0 0 0.998 0.002 Obs170 1 1 0.051 0.949 Obs171 0 0 0.65 0.35 Obs172 1 1 0 1 Obs173 0 0 0.58 0.42 Obs174 1 1 0.046 0.954 Obs175 1 1 0.001 0.999 Obs176 1 1 0 1 Obs177 1 1 0.001 0.999 Obs178 1 1 0.1 0.9 Obs179 1 1 0.013 0.987 Obs180 0 0 0.989 0.011 Obs181 1 1 0.352 0.648 Obs182 1 1 0.014 0.986 Obs183 1 1 0.066 0.934 Obs184 1 1 0.054 0.946 Obs185 1 1 0.001 0.999 Obs186 1 1 0.005 0.995 Obs187 1 1 0.03 0.97 Obs188 1 1 0.011 0.989 Obs189 1 1 0.144 0.856 Obs190 1 1 0.116 0.884 Obs191 1 0 0.837 0.163 Obs192 0 1 0.116 0.884 Obs193 0 0 0.719 0.281 Obs194 0 0 0.984 0.016 Obs195 1 1 0 1 Obs196 0 0 0.981 0.019 Obs197 1 1 0.073 0.927 Obs198 1 1 0.208 0.792 Obs199 1 1 0.026 0.974 Obs200 1 1 0 1 Obs201 0 0 0.926 0.074 Obs202 1 1 0.017 0.983 Obs203 0 0 0.521 0.479 Obs204 1 1 0.003 0.997 Obs205 1 1 0.232 0.768 Obs206 1 1 0.16 0.84 Obs207 1 1 0.001 0.999 Obs208 0 0 0.984 0.016 Obs209 1 1 0.03 0.97 Obs210 0 0 0.919 0.081 Obs211 0 0 0.537 0.463 Obs212 1 1 0.005 0.995 Obs213 1 1 0.054 0.946 Obs214 1 1 0.16 0.84 Obs215 0 0 0.756 0.244 Obs216 1 1 0.124 0.876 Obs217 0 1 0.434 0.566 Obs218 1 1 0.232 0.768 Obs219 0 0 0.789 0.211 Obs220 1 1 0.36 0.64 Obs221 0 0 0.882 0.118 Obs222 0 0 0.865 0.135 Obs223 1 1 0.232 0.768 Obs224 1 1 0.005 0.995 Obs225 1 1 0 1 Obs226 1 1 0.136 0.864 Obs227 0 0 0.926 0.074 Obs228 0 0 0.725 0.275 Obs229 1 1 0.207 0.793 Obs230 1 1 0.031 0.969 Obs231 0 0 0.521 0.479 Obs232 0 0 0.537 0.463 Obs233 0 0 0.865 0.135 Obs234 1 0 0.621 0.379 Obs235 1 1 0.005 0.995 Obs236 0 0 0.895 0.105 Obs237 0 0 0.962 0.038 Obs238 1 1 0.003 0.997 Obs239 0 0 0.914 0.086 Obs240 0 0 0.537 0.463 Obs241 1 0 0.926 0.074 Obs242 1 1 0.051 0.949 Obs243 1 1 0.073 0.927 Obs244 1 0 0.606 0.394 Obs245 1 1 0.003 0.997 Obs246 1 1 0 1 Obs247 1 1 0 1 Obs248 1 1 0.051 0.949 Obs249 0 1 0.213 0.787 Obs250 1 1 0.221 0.779 Obs251 0 0 0.643 0.357 Obs252 0 0 0.65 0.35 Obs253 0 0 0.58 0.42 Obs254 1 1 0.046 0.954 Obs255 0 0 0.621 0.379 Obs256 0 1 0.011 0.989 Obs257 1 1 0.016 0.984 Obs258 0 0 0.946 0.054 Obs259 1 1 0.277 0.723 Obs260 1 1 0.002 0.998 Obs261 0 0 0.989 0.011 Obs262 1 1 0.005 0.995 Obs263 0 0 0.58 0.42 Obs264 1 1 0.001 0.999 Obs265 0 0 0.882 0.118 Obs266 0 0 0.882 0.118 Obs267 0 0 0.901 0.099 Obs268 1 1 0.345 0.655 Obs269 0 1 0.106 0.894 Obs270 0 1 0.36 0.64 Obs271 0 0 0.968 0.032 Obs272 1 1 0.016 0.984 Obs273 1 1 0 1 Obs274 1 1 0.221 0.779 Obs275 0 0 0.841 0.159 Obs276 1 1 0.466 0.534 Obs277 1 1 0.001 0.999 Obs278 0 1 0.352 0.648 Obs279 1 1 0.002 0.998 Obs280 1 1 0.013 0.987 Obs281 0 0 0.919 0.081 Obs282 0 0 0.968 0.032 Obs283 0 0 0.841 0.159 Obs284 0 1 0.32 0.68 Obs285 1 1 0.016 0.984 Obs286 1 1 0.003 0.997 Obs287 0 0 0.946 0.054 Obs288 1 1 0.002 0.998 Obs289 1 1 0.031 0.969 Obs290 1 0 0.756 0.244 Obs291 1 1 0.003 0.997 Obs292 1 1 0.32 0.68 Obs293 1 1 0.073 0.927 Obs294 0 0 0.895 0.105 Obs295 1 1 0.001 0.999 Obs296 1 1 0.007 0.993 Obs297 1 1 0.026 0.974 Obs298 1 1 0.033 0.967 Obs299 1 1 0.008 0.992 Obs300 1 1 0 1 Obs301 0 0 0.561 0.439 Obs302 0 0 0.846 0.154 Obs303 0 0 0.719 0.281 Obs304 0 1 0.408 0.592 Obs305 0 1 0.208 0.792 Obs306 1 1 0.003 0.997 Obs307 1 1 0.016 0.984 Obs308 1 1 0.001 0.999 Obs309 1 1 0 1 Obs310 1 1 0.003 0.997 Obs311 0 0 0.561 0.439 Obs312 0 0 0.901 0.099 Obs313 1 1 0.054 0.946 Obs314 1 1 0 1 Obs315 0 0 0.926 0.074 Obs316 0 1 0.156 0.844 Obs317 1 1 0.051 0.949 Obs318 1 1 0 1 Obs319 0 0 0.58 0.42 Obs320 0 0 0.914 0.086 Obs321 1 1 0.408 0.592 Obs322 1 1 0.082 0.918 Obs323 1 1 0.004 0.996 Obs324 1 1 0 1 Obs325 1 1 0.004 0.996 Obs326 0 1 0.136 0.864 Obs327 0 0 0.521 0.479 Obs328 1 1 0 1 Obs329 1 1 0.345 0.655 Obs330 1 1 0.008 0.992 Obs331 1 1 0.054 0.946 Obs332 1 1 0.016 0.984 Obs333 1 1 0.073 0.927 Obs334 0 1 0.136 0.864 Obs335 1 1 0.016 0.984 Obs336 1 0 0.626 0.374 Obs337 0 0 0.962 0.038 Obs338 0 0 0.901 0.099 Obs339 1 1 0.232 0.768 Obs340 0 0 0.65 0.35 Obs341 1 1 0.156 0.844 Obs342 0 0 0.789 0.211 Obs343 1 1 0.051 0.949 Obs344 1 1 0.003 0.997 Obs345 1 1 0.002 0.998 Obs346 1 1 0.144 0.856 Obs347 0 0 0.65 0.35 Obs348 0 0 0.719 0.281 Obs349 1 1 0.156 0.844 Obs350 0 1 0.116 0.884 Obs351 1 1 0.002 0.998 Obs352 1 1 0.434 0.566 Obs353 1 1 0.207 0.793 Obs354 0 0 0.561 0.439 Obs355 1 0 0.882 0.118 Obs356 1 1 0.03 0.97 Obs357 1 1 0.001 0.999 Obs358 1 1 0.207 0.793 Obs359 0 1 0.156 0.844 Obs360 1 1 0.002 0.998 Obs361 1 1 0.007 0.993 Obs362 1 1 0.007 0.993 Obs363 1 1 0.001 0.999 Obs364 1 1 0.028 0.972 Obs365 1 1 0.005 0.995 Obs366 0 0 0.919 0.081 Obs367 1 1 0.005 0.995 Obs368 1 1 0.136 0.864 Obs369 1 1 0.019 0.981 Obs370 1 1 0 1 Obs371 0 0 0.989 0.011 Obs372 1 1 0.005 0.995 Obs373 0 0 0.537 0.463 Obs374 0 0 0.895 0.105 Obs375 1 1 0.277 0.723 Obs376 0 0 0.984 0.016 Obs377 1 1 0.106 0.894 Obs378 1 1 0.208 0.792 Obs379 1 0 0.719 0.281 Obs380 1 1 0.004 0.996 Obs381 1 1 0.066 0.934 Obs382 1 1 0.1 0.9 Obs383 0 0 0.725 0.275 Obs384 1 1 0.116 0.884 Obs385 1 1 0.116 0.884 Obs386 1 1 0.003 0.997 Obs387 1 1 0.004 0.996 Obs388 1 1 0.001 0.999 Obs389 1 1 0 1 Obs390 1 1 0.026 0.974 Obs391 0 0 0.606 0.394 Obs392 1 1 0 1 Obs393 1 1 0.116 0.884 Obs394 1 1 0.434 0.566 Obs395 1 1 0.073 0.927 Obs396 0 0 0.963 0.037 Obs397 1 0 0.837 0.163 Obs398 1 1 0.008 0.992 Obs399 1 1 0.124 0.876 Obs400 0 0 0.895 0.105 Obs401 1 1 0.073 0.927 Obs402 0 1 0.208 0.792 Obs403 1 0 0.606 0.394 Obs404 1 1 0.466 0.534 Obs405 1 1 0.082 0.918 Obs406 1 1 0.003 0.997 Obs407 1 1 0.001 0.999 Obs408 1 0 0.865 0.135 Obs409 1 1 0.026 0.974 Obs410 1 1 0.03 0.97 Obs411 1 1 0 1 Obs412 1 1 0.008 0.992 Obs413 1 1 0 1 Obs414 1 1 0 1 Obs415 1 1 0 1 Obs416 0 1 0.213 0.787 Obs417 1 1 0.481 0.519 Obs418 1 1 0.031 0.969 Obs419 0 0 0.997 0.003 Obs420 0 1 0.213 0.787 Obs421 1 1 0.002 0.998 Obs422 0 0 0.626 0.374 Obs423 1 1 0.136 0.864 Obs424 1 1 0.073 0.927 Obs425 1 1 0.001 0.999 Obs426 1 0 0.719 0.281 Obs427 0 0 0.895 0.105 Obs428 0 0 0.841 0.159 Obs429 1 1 0.017 0.983 Obs430 1 1 0 1 Obs431 1 1 0.051 0.949 Obs432 1 1 0.003 0.997 Obs433 0 0 0.882 0.118 Obs434 0 0 0.725 0.275 Obs435 1 1 0.038 0.962 Obs436 1 1 0 1 Obs437 1 1 0.038 0.962 Obs438 1 1 0.1 0.9 Obs439 0 0 0.719 0.281 Obs440 0 1 0.116 0.884 Obs441 0 0 0.942 0.058 Obs442 1 1 0.066 0.934 Obs443 1 1 0.007 0.993 Obs444 1 1 0.32 0.68 Obs445 1 1 0.345 0.655 Obs446 0 1 0.36 0.64 Obs447 1 1 0.007 0.993 Obs448 1 1 0.221 0.779 Obs449 1 1 0.144 0.856 Obs450 0 0 0.962 0.038 Obs451 1 1 0.146 0.854 Obs452 0 0 0.997 0.003 Obs453 0 1 0.352 0.648 Obs454 1 1 0.013 0.987 Obs455 0 0 0.963 0.037 Obs456 1 1 0.156 0.844 Obs457 1 1 0.002 0.998 Obs458 1 1 0.038 0.962 Obs459 1 1 0 1 Obs460 1 1 0.001 0.999 Obs461 1 1 0.005 0.995 Obs462 0 0 0.963 0.037 Obs463 1 1 0.017 0.983 Obs464 1 1 0.002 0.998 Obs465 1 0 0.756 0.244 Obs466 0 0 0.725 0.275 Obs467 1 1 0.019 0.981 Obs468 0 0 0.914 0.086 Obs469 1 1 0.028 0.972 Obs470 1 1 0.051 0.949 Obs471 0 0 0.789 0.211 Obs472 1 1 0.066 0.934 Obs473 1 1 0.051 0.949 Obs474 1 0 0.756 0.244 Obs475 0 0 0.643 0.357 Obs476 1 1 0.073 0.927 Obs477 1 1 0.124 0.876 Obs478 1 1 0.003 0.997 Obs479 1 1 0.146 0.854 Obs480 1 1 0.466 0.534 Obs481 1 1 0.136 0.864 Obs482 1 1 0 1 Obs483 1 1 0.005 0.995 Obs484 0 0 0.997 0.003 Obs485 1 1 0.008 0.992 Obs486 1 1 0.017 0.983 Obs487 1 1 0.32 0.68 Obs488 1 1 0.046 0.954 Obs489 0 1 0.073 0.927 Obs490 1 1 0.002 0.998 Obs491 1 1 0.046 0.954 Obs492 1 1 0.082 0.918 Obs493 1 1 0.004 0.996 Obs494 0 0 0.998 0.002 Obs495 1 1 0.001 0.999 Obs496 0 0 0.998 0.002 Obs497 1 0 0.621 0.379 Obs498 0 0 0.841 0.159 Obs499 0 0 0.789 0.211 Obs500 0 0 0.626 0.374 Obs501 1 1 0.016 0.984 Obs502 1 1 0.073 0.927 Obs503 1 1 0.014 0.986 Obs504 1 1 0.408 0.592 Obs505 1 1 0.019 0.981 Obs506 1 1 0 1 Obs507 1 1 0.033 0.967 Obs508 0 0 0.626 0.374 Obs509 0 0 0.997 0.003 Obs510 1 1 0.013 0.987 Obs511 1 1 0.16 0.84 Obs512 1 1 0 1 Obs513 1 1 0 1 Obs514 1 1 0.408 0.592 Obs515 0 0 0.846 0.154 Obs516 0 0 0.946 0.054 Obs517 0 1 0.352 0.648 Obs518 0 0 0.963 0.037 Obs519 0 1 0.213 0.787 Obs520 1 1 0 1 Obs521 0 0 0.846 0.154 Obs522 1 0 0.561 0.439 Obs523 1 1 0.466 0.534 Obs524 1 1 0 1 Obs525 1 1 0.004 0.996 Obs526 0 1 0.106 0.894 Obs527 0 0 0.846 0.154 Obs528 1 1 0.001 0.999 Obs529 1 1 0.016 0.984 Obs530 1 1 0.277 0.723 Obs531 0 0 0.561 0.439 Obs532 0 0 0.719 0.281 Obs533 1 1 0.144 0.856 Obs534 1 1 0.001 0.999 Obs535 1 1 0 1 Obs536 1 1 0.019 0.981 Obs537 1 1 0.146 0.854 Obs538 1 1 0.051 0.949 Obs539 1 1 0 1 Obs540 1 1 0.36 0.64

TABLE 3B Goodness of fit statistics (variable BC) Statistic Independent Full Observations 432 432 Sum of weights 432 432 DF 431 428 −2 Log(Likelihood) 542.755 240.107 2 R(McFadden) 0 0.558 2 R(Cox and Snell) 0 0.504 2 R(Nagelkerke) 0 0.701 AIC 544.755 248.107 SBC 548.823 264.381 Iterations 0 6

TABLE 3C Test of null hypothesis Pr(BC = 1) = 0.678 Statistic DF Chi-square 2 Pr > Chi −2 Log(Likelihood) 3 302.647 <0.0001 Score 3 204.837 <0.0001 Wald 3 94.786 <0.0001

TABLE 3D Type-II analysis (variable BC) Chi-square Chi-square Source DF (Wald) Pr > Wald (LR) Pr > LR BS 1 49.62 <0.0001 72.563 <0.0001 BP 1 68.961 <0.0001 117.499 <0.0001 SCS 1 75.014 <0.0001 192.392 <0.0001

TABLE 3E Hosmer-Lemeshow test (variable BC0) Statistic Chi-square DF 2 Pr > Chi Hosmer-Lemeshow Statistic 8.73 8 0.366

TABLE 3F Model parameters (variable BC) Odd Odds Wald Wald ratio ratio Wald Lower Upper lower Upper Standard Chi- Pr > bound bound Odds bounds bound Source Value error Square 2 Chi (95%) (95%) ratio (95%) (95%) Intercept −3.235 0.594 29.614 <0.0001 −4.400 −2.070 BS −0.633 0.09 49.62 <0.0001 −0.809 −0.457 0.531 0.445 0.633 BP 0.091 0.011 68.961 <0.0001 0.07 0.113 1.095 1.072 1.119 SCS 0.01 0.001 75.014 <0.0001 0.007 0.012 1.01 1.007 1.012

TABLE 3G Standardized coefficients (variable BC) Wald Wald Wald Lower Upper Standard Chi- Pr > bound bound Source Value error Square 2 Chi (95%) (95%) BS −0.759 0.108 49.62 <0.0001 −0.970 −0.548 BP 1.031 0.124 68.961 <0.0001 0.787 1.274 SCS 1.613 0.186 75.014 <0.0001 1.248 1.978

TABLE 3H Classification for the training samples (variable BC) to from 0 1 Total % Correct 0 116 23 139 83.45% 1 23 270 293 92.15% Total 139 293 432 89.35%

TABLE 3I Classification for the validation sample (variable BC) to from 0 1 Total % Correct 0 27 7 34 79.41% 1 4 70 74 94.59% Total 31 77 108 89.81%

2 2 2 2 The digital samples generated in the present disclosure are limited by computational constraints. The focus was to get the smallest SCS as close as possible to core-plug and rotary side wall core dimensions; hence a SCS of about 25 cmwas achieved. This is equivalent to a cylindrical sample with 2.2-inch diameter, close to the 1 to 1.5 inches of typical core plugs. Similarly, the 100 cmSCS has an area comparable to a cylinder with 4.4-inch diameter. This is similar in SCS to 4-inch drill core. However, the digital samples of the present disclosure have a 1-meter-length which is substantially longer than the samples used in the laboratory to measure permeability. The typical length of laboratory samples depends on their diameter. For example, core plugs typically are only 1 to 2 inches long. Whole core (full diameter) analyses typically are somewhere in the range of about 4 to 10-inches. Results of the experimentation conducted in the present disclosure allows for analysis of the controls on the shorter-than-1 m-samples. In order to perform the said procedure, the results from 25 cmand 100 cmSCS runs were analysed as proxies for core plugs and 4-inch core, respectively, documented in Table 5A and Table 6A. Out of these 180 samples (90 samples for each cross section), 67 samples have LCBV that span the entire 1 m length of the digital samples, and the rest 113 samples have LCBV with lengths less than 1 m. For the 67 samples with 1 m-long LCBVs, the combination of SCS, BP and BS yield results that assure that any length of vertical sample will represent the vertical connectivity of the burrows. For the other 113 samples, however, the combination of SCS, BP and BS do not yield LCBVs that span the entire 1-meter-length, and thus, shorter samples cannot be guaranteed of representing the vertical connectivity of the burrows. For these, the probability of a particular 1- to 10-inch-long sample falling within the LCBV depends on the SCS, BS, BP, and the length of the sample segment. The probability of those sample segments falling fully within the LCBV, and thus correctly representing permeability, can be calculated as follows. Results provided the length of the LCBV for each combination of SCS, BS, and BP, the results are provided in Table 4A. Given samples of specified length (lS), a number of samples can be calculated (#Sa) that may fall within the length of the LCBV as represented in equation 2. Since it is unlikely that the two samples at the top and bottom of the LCBV lie fully within the LCBV, the integer of 2 is subtracted from the product of equation 2 to produce the number of samples of a specified length that would fall fully within the length of the LCBV, as represented by equation 3. Since the columns have 1-m length, the total number of samples possible (#St), of a specified length (lS) is produced by dividing 100 by the length of the subsample (lS in cm), as represented in equation 4. The probability of having the LCBV represented in the subsamples may be calculated by dividing the product of equation 3 by the product of equation 4 as represented in equation 5.

Where, #Sa is total samples possible in LCBV, #Sb is total samples possible fully in LCBV, #St is total samples possible of specified length, lLCBV is length of the LCBV, lS is length of the sample, and pLCBV is the probability of having the LCBV represented in a sample.

2 11 FIG.C 11 FIG.A 11 FIG.B Further, a total of 71 digital samples has LCBV less than 100 cm in the digital samples meant to simulate core plug sampling, with SCS of 25 cm(90 samples), as documented in Table 4A. Applying the above equations to 71 digital samples using the lS of 5 cm, calculates the probability of a 5 cm-long sample sampling the LCBV (represented by pLCBV). As shown in, only 28% of digital samples have 0.5 or greater pLCBV. The same analysis was carried out for shorter sample lengths (1 and 2.5 cm). For the 1 cm length samples, as shown in, only 37% of digital samples yield 0.5 or greater pLCBV. For the 2.5 cm length samples, as shown in, 34% of digital samples yield 0.5 or greater pLCBV.

2 12 12 FIGS.A-C 12 FIG.A 12 FIG.B 12 FIG.C Similarly, for the digital samples meant to simulate whole core, with SCS of 100 cm(90 samples), as documented in Table 6A, 42 of the digital samples have LCBV of less than 100 cm. Applying the above specified equations to the 42 samples, with lS of 10 cm, 20 cm, and 25 cm to simulate various lengths of whole-core samples may be simulated, as shown in. The calculated pLCBV of these samples ranges from 0% to 78% for the 10 cm length, from 0% to 58% for the 20 cm length, and from 0% to 48% for the 25 cm length. As shown in, for 10-cm long samples, only 33% of digital samples have 0.5 or greater pLCBV. As shown in, for 20-cm long samples, only 12% of digital samples have 0.5 or greater pLCBV. As shown in, for 25-cm long samples, none of the digital samples have 0.5 or greater pLCBV.

11 11 FIGS.A-C 12 12 FIGS.A-C The results of this process may be employed for users designing sampling routines. They enter their BP, BS, and SCS into the logistic regression algorithm. That algorithm produces output of if burrows represent the connection of burrows across the 1 m model, for a particular SCS. If that burrow connectivity is represented, then any sample length with that SCS should represent the rock's burrow permeability appropriately. For the many SCS samples, where the output does not show 100% probability of burrows connecting across the 1 m sample, the user can take the additional step derived from the relationships outlined inand. To apply this, users may employ look-up Tables 5A-5D, and look-up Tables 6A-6D, choosing core plug or whole core SCS. They then must choose a sample length. This produces a probability of a sample, of that specified width and length, falling within the LCBV and correctly representing the permeability.

TABLE 4A Length of LCBV for each combination of BS, BP, SCS BS BP SCS Length BS BP SCS Length 9 25 5 16 2.6 50 15 100 9 25 5 18 2.6 75 15 100 9 50 5 19 2.6 25 15 100 9 75 5 23 2.6 50 15 100 9 50 5 24 2.6 75 15 100 9 50 5 25 2.6 25 15 100 9 25 5 34 2.6 50 15 100 9 75 5 35 2.6 75 15 100 9 50 5 37 9 25 20 19 9 75 5 100 9 25 20 31 9 25 5 35 9 25 20 35 9 50 5 61 9 25 20 40 9 25 5 62 9 25 20 60 9 75 5 71 9 25 20 66 9 75 5 100 9 50 20 70 5 50 5 19 9 50 20 88 5.1 25 5 25 9 75 20 100 5.1 50 5 33 9 75 20 100 5.1 25 5 38 9 50 20 100 5.1 50 5 39 9 75 20 100 5.1 75 5 44 9 50 20 100 5.1 75 5 49 9 75 20 100 5.1 25 5 51 9 50 20 100 5.1 75 5 58 9 75 20 100 4 25 5 15 9 50 20 100 4 25 5 18 9 75 20 100 4 25 5 24 5.1 25 20 40 4 50 5 25 5.1 25 20 60 4 25 5 28 5.1 25 20 66 4 50 5 33 5.1 25 20 71 4 25 5 35 5.1 25 20 80 4 50 5 36 5.1 50 20 100 4 50 5 49 5.1 75 20 100 4 50 5 58 5.1 50 20 100 4 75 5 90 5.1 75 20 100 4 75 5 92 5.1 50 20 100 4 75 5 100 5.1 75 20 100 3.4 50 5 24 5.1 50 20 100 3.4 25 5 28 5.1 75 20 100 3.4 25 5 32 5.1 50 20 100 3.4 25 5 32 5.1 75 20 100 3.4 50 5 36 5.1 25 20 100 3.4 50 5 46 5.1 50 20 100 3.4 50 5 56 5.1 75 20 100 3.4 25 5 69 4 25 20 61 3.4 75 5 100 4 25 20 89 3.4 75 5 100 4 25 20 100 3.4 75 5 100 4 50 20 100 3.4 75 5 100 4 75 20 100 2.7 25 5 13 4 25 20 100 2.7 25 5 27 4 50 20 100 2.7 50 5 45 4 75 20 100 2.7 25 5 66 4 25 20 100 2.7 50 5 100 4 50 20 100 2.7 50 5 100 4 75 20 100 2.7 25 5 71 4 50 20 100 2.7 25 5 44 4 75 20 100 2.7 50 5 100 4 50 20 100 2.7 50 5 100 4 75 20 100 2.7 75 5 65 4 25 20 100 2.7 75 5 92 4 50 20 100 2.7 75 5 100 4 75 20 100 2.7 75 5 100 3.4 25 20 100 2.7 75 5 100 3.4 50 20 100 2.6 25 5 23 3.4 75 20 100 2.6 25 5 24 3.4 25 20 100 2.6 25 5 28 3.4 50 20 100 2.6 50 5 35 3.4 75 20 100 2.6 25 5 40 3.4 25 20 100 2.6 50 5 52 3.4 50 20 100 2.6 50 5 66 3.4 75 20 100 2.6 50 5 68 3.4 25 20 100 2.6 75 5 75 3.4 50 20 100 2.6 75 5 80 3.4 75 20 100 2.6 75 5 100 3.4 25 20 100 2.6 75 5 100 3.4 50 20 100 5.1 25 5 27 3.4 75 20 100 5.1 25 5 33 2.7 25 20 100 5.1 50 5 36 2.7 50 20 100 5.1 75 5 72 2.7 75 20 100 5.1 75 5 77 2.7 25 20 100 5.1 50 5 92 2.7 50 20 100 4 75 5 52 2.7 75 20 100 4 75 5 60 2.7 25 20 100 3.4 25 5 72 2.7 50 20 100 3.4 50 5 100 2.7 75 20 100 3.4 75 5 100 2.7 25 20 100 2.6 25 5 57 2.7 50 20 100 2.6 50 5 79 2.7 75 20 100 2.6 75 5 100 2.7 25 20 100 9 25 10 16 2.7 50 20 100 9 25 10 19 2.7 75 20 100 9 50 10 25 2.6 25 20 100 9 25 10 34 2.6 50 20 100 9 25 10 36 2.6 75 20 100 9 25 10 54 2.6 25 20 100 9 50 10 72 2.6 50 20 100 9 50 10 73 2.6 75 20 100 9 50 10 85 2.6 25 20 100 9 50 10 90 2.6 50 20 100 9 75 10 100 2.6 75 20 100 9 75 10 100 2.6 25 20 100 9 75 10 100 2.6 50 20 100 9 75 10 100 2.6 75 20 100 9 75 10 100 2.6 25 20 100 5.1 25 10 24 2.6 50 20 100 5.1 25 10 26 2.6 75 20 100 5.1 50 10 32 9 50 25 4 5.1 50 10 34 9 25 25 40 5.1 50 10 42 9 25 25 51 5.1 25 10 43 9 25 25 67 5.1 25 10 48 9 25 25 72 5.1 50 10 53 9 75 25 100 5.1 25 10 64 9 50 25 100 5.1 50 10 84 9 75 25 100 5.1 75 10 84 9 50 25 100 5.1 75 10 100 9 75 25 100 5.1 75 10 100 9 50 25 100 5.1 75 10 100 9 75 25 100 5.1 75 10 100 5.1 25 25 76 4 25 10 37 5.1 25 25 83 4 25 10 42 5.1 25 25 100 4 25 10 47 5.1 50 25 100 4 25 10 49 5.1 75 25 100 4 50 10 60 5.1 25 25 100 4 50 10 60 5.1 50 25 100 4 50 10 64 5.1 75 25 100 4 50 10 74 5.1 50 25 100 4 25 10 100 5.1 75 25 100 4 75 10 100 5.1 50 25 100 4 75 10 100 5.1 75 25 100 4 50 10 100 4 25 25 51 4 75 10 100 4 25 25 100 4 75 10 100 4 50 25 100 4 75 10 100 4 75 25 100 3.4 25 10 49 4 25 25 100 3.4 25 10 49 4 50 25 100 3.4 25 10 64 4 75 25 100 3.4 25 10 96 4 50 25 100 3.4 50 10 100 4 75 25 100 3.4 75 10 100 4 25 25 100 3.4 50 10 100 4 50 25 100 3.4 75 10 100 4 75 25 100 3.4 50 10 100 3.4 25 25 100 3.4 75 10 100 3.4 50 25 100 3.4 50 10 100 3.4 75 25 100 3.4 75 10 100 3.4 25 25 100 3.4 25 10 100 3.4 50 25 100 3.4 50 10 100 3.4 75 25 100 3.4 75 10 100 3.4 25 25 100 2.7 25 10 80 3.4 50 25 100 2.7 25 10 50 3.4 75 25 100 2.7 50 10 64 3.4 25 25 100 2.7 25 10 98 3.4 50 25 100 2.7 25 10 96 3.4 75 25 100 2.7 50 10 100 2.7 25 25 100 2.7 50 10 76 2.7 50 25 100 2.7 25 10 100 2.7 75 25 100 2.7 50 10 100 2.7 25 25 100 2.7 75 10 100 2.7 50 25 100 2.7 75 10 100 2.7 75 25 100 2.7 75 10 100 2.7 25 25 100 2.7 50 10 100 2.7 50 25 100 2.7 75 10 100 2.7 75 25 100 2.7 75 10 100 2.7 25 25 100 2.6 25 10 67 2.7 50 25 100 2.6 25 10 80 2.7 75 25 100 2.6 25 10 91 2.6 25 25 100 2.6 25 10 100 2.6 50 25 100 2.6 50 10 100 2.6 75 25 100 2.6 75 10 100 2.6 25 25 100 2.6 50 10 100 2.6 50 25 100 2.6 75 10 100 2.6 75 25 100 2.6 25 10 100 2.6 25 25 100 2.6 50 10 100 2.6 50 25 100 2.6 75 10 100 2.6 75 25 100 2.6 50 10 100 2.6 25 25 100 2.6 75 10 100 2.6 50 25 100 2.6 50 10 100 2.6 75 25 100 2.6 75 10 100 9 25 30 15 9 25 15 22 9 50 30 30 9 25 15 41 9 25 30 66 9 25 15 49 9 25 30 100 9 25 15 65 9 50 30 100 9 50 15 74 9 75 30 100 9 50 15 82 9 50 30 100 9 50 15 100 9 75 30 100 9 75 15 100 9 25 30 100 9 75 15 100 9 50 30 100 9 75 15 100 9 75 30 100 9 50 15 100 9 75 30 100 9 75 15 100 5.1 25 30 71 5.1 25 15 30 5.1 25 30 100 5.1 25 15 39 5.1 50 30 100 5.1 25 15 59 5.1 75 30 100 5.1 50 15 78 5.1 25 30 100 5.1 75 15 79 5.1 50 30 100 5.1 50 15 95 5.1 75 30 100 5.1 50 15 100 5.1 50 30 100 5.1 75 15 100 5.1 75 30 100 5.1 50 15 100 5.1 25 30 100 5.1 75 15 100 5.1 50 30 100 5.1 25 15 100 5.1 75 30 100 5.1 75 15 100 4 25 30 100 4 25 15 40 4 50 30 100 4 75 15 48 4 75 30 100 4 25 15 51 4 25 30 100 4 25 15 82 4 50 30 100 4 25 15 84 4 75 30 100 4 50 15 100 4 25 30 100 4 50 15 100 4 50 30 100 4 75 15 100 4 75 30 100 4 50 15 100 4 25 30 100 4 75 15 100 4 50 30 100 4 50 15 100 4 75 30 100 4 75 15 100 3.4 25 30 100 3.4 25 15 54 3.4 50 30 100 3.4 25 15 64 3.4 75 30 100 3.4 25 15 99 3.4 25 30 100 3.4 50 15 100 3.4 50 30 100 3.4 75 15 100 3.4 75 30 100 3.4 25 15 100 3.4 25 30 100 3.4 50 15 100 3.4 50 30 100 3.4 75 15 100 3.4 75 30 100 3.4 50 15 100 3.4 25 30 100 3.4 75 15 100 3.4 50 30 100 3.4 50 15 100 3.4 75 30 100 3.4 75 15 100 2.7 25 30 100 3.4 25 15 100 2.7 50 30 100 3.4 50 15 100 2.7 75 30 100 3.4 75 15 100 2.7 25 30 100 2.7 25 15 100 2.7 50 30 100 2.7 50 15 100 2.7 75 30 100 2.7 75 15 100 2.7 25 30 100 2.7 25 15 100 2.7 50 30 100 2.7 50 15 100 2.7 75 30 100 2.7 75 15 100 2.7 25 30 100 2.7 25 15 100 2.7 50 30 100 2.7 50 15 100 2.7 75 30 100 2.7 75 15 100 2.6 25 30 100 2.7 25 15 100 2.6 50 30 100 2.7 50 15 100 2.6 75 30 100 2.7 75 15 100 2.6 25 30 100 2.7 25 15 100 2.6 50 30 100 2.7 50 15 100 2.6 75 30 100 2.7 75 15 100 2.6 25 30 100 2.6 25 15 100 2.6 50 30 100 2.6 50 15 100 2.6 75 30 100 2.6 75 15 100 2.6 25 30 100 2.6 25 15 100 2.6 50 30 100 2.6 50 15 100 2.6 75 30 100 2.6 75 15 100 2.6 25 15 100

TABLE 4B Burrow connectivity for each combination of BS, BP, SCS Burrow Burrow BS BP SCS connectivity BS BP SCS connectivity 2.6 25 5 0 4 25 5 0 2.6 25 5 0 4 25 5 0 2.6 25 5 0 4 25 5 0 2.6 25 5 0 4 25 5 0 2.6 25 5 0 4 25 5 0 2.6 25 10 0 4 25 10 0 2.6 25 10 0 4 25 10 0 2.6 25 10 0 4 25 10 0 2.6 25 10 1 4 25 10 1 2.6 25 10 1 4 25 10 1 2.6 25 15 0 4 25 15 0 2.6 25 15 1 4 25 15 0 2.6 25 15 1 4 25 15 0 2.6 25 15 1 4 25 15 0 2.6 25 15 1 4 25 15 1 2.6 25 20 1 4 25 20 0 2.6 25 20 1 4 25 20 1 2.6 25 20 1 4 25 20 1 2.6 25 20 1 4 25 20 1 2.6 25 20 1 4 25 20 1 2.6 25 25 1 4 25 25 0 2.6 25 25 1 4 25 25 1 2.6 25 25 1 4 25 25 1 2.6 25 25 1 4 25 25 1 2.6 25 25 1 4 25 25 1 2.6 25 30 1 4 25 30 1 2.6 25 30 1 4 25 30 1 2.6 25 30 1 4 25 30 1 2.6 25 30 1 4 25 30 1 2.6 25 30 1 4 25 30 1 2.6 50 5 0 4 50 5 0 2.6 50 5 0 4 50 5 0 2.6 50 5 0 4 50 5 0 2.6 50 5 0 4 50 5 0 2.6 50 5 0 4 50 5 0 2.6 50 10 1 4 50 10 0 2.6 50 10 1 4 50 10 0 2.6 50 10 1 4 50 10 0 2.6 50 10 1 4 50 10 0 2.6 50 10 1 4 50 10 1 2.6 50 15 1 4 50 15 1 2.6 50 15 1 4 50 15 1 2.6 50 15 1 4 50 15 1 2.6 50 15 1 4 50 15 1 2.6 50 15 1 4 50 15 1 2.6 50 20 1 4 50 20 1 2.6 50 20 1 4 50 20 1 2.6 50 20 1 4 50 20 1 2.6 50 20 1 4 50 20 1 2.6 50 20 1 4 50 20 1 2.6 50 25 1 4 50 25 1 2.6 50 25 1 4 50 25 1 2.6 50 25 1 4 50 25 1 2.6 50 25 1 4 50 25 1 2.6 50 25 1 4 50 25 1 2.6 50 30 1 4 50 30 1 2.6 50 30 1 4 50 30 1 2.6 50 30 1 4 50 30 1 2.6 50 30 1 4 50 30 1 2.6 50 30 1 4 50 30 1 2.6 75 5 1 4 75 5 0 2.6 75 5 1 4 75 5 0 2.6 75 5 1 4 75 5 0 2.6 75 5 1 4 75 5 0 2.6 75 5 1 4 75 5 1 2.6 75 10 1 4 75 10 1 2.6 75 10 1 4 75 10 1 2.6 75 10 1 4 75 10 1 2.6 75 10 1 4 75 10 1 2.6 75 10 1 4 75 10 1 2.6 75 15 1 4 75 15 1 2.6 75 15 1 4 75 15 1 2.6 75 15 1 4 75 15 1 2.6 75 15 1 4 75 15 1 2.6 75 15 1 4 75 15 1 2.6 75 20 1 4 75 20 1 2.6 75 20 1 4 75 20 1 2.6 75 20 1 4 75 20 1 2.6 75 20 1 4 75 20 1 2.6 75 20 1 4 75 20 1 2.6 75 25 1 4 75 25 1 2.6 75 25 1 4 75 25 1 2.6 75 25 1 4 75 25 1 2.6 75 25 1 4 75 25 1 2.6 75 25 1 4 75 25 1 2.6 75 30 1 4 75 30 1 2.6 75 30 1 4 75 30 1 2.6 75 30 1 4 75 30 1 2.6 75 30 1 4 75 30 1 2.6 75 30 1 4 75 30 1 2.7 25 5 0 5.1 25 5 0 2.7 25 5 0 5.1 25 5 0 2.7 25 5 0 5.1 25 5 0 2.7 25 5 0 5.1 25 5 0 2.7 25 5 0 5.1 25 5 0 2.7 25 10 0 5.1 25 10 0 2.7 25 10 0 5.1 25 10 0 2.7 25 10 0 5.1 25 10 0 2.7 25 10 0 5.1 25 10 0 2.7 25 10 0 5.1 25 10 0 2.7 25 15 0 5.1 25 15 0 2.7 25 15 1 5.1 25 15 0 2.7 25 15 1 5.1 25 15 0 2.7 25 15 1 5.1 25 15 0 2.7 25 15 1 5.1 25 15 1 2.7 25 20 1 5.1 25 20 0 2.7 25 20 1 5.1 25 20 0 2.7 25 20 1 5.1 25 20 0 2.7 25 20 1 5.1 25 20 0 2.7 25 20 1 5.1 25 20 1 2.7 25 25 1 5.1 25 25 0 2.7 25 25 1 5.1 25 25 0 2.7 25 25 1 5.1 25 25 1 2.7 25 25 1 5.1 25 25 1 2.7 25 25 1 5.1 25 25 1 2.7 25 30 1 5.1 25 30 0 2.7 25 30 1 5.1 25 30 1 2.7 25 30 1 5.1 25 30 1 2.7 25 30 1 5.1 25 30 1 2.7 25 30 1 5.1 25 30 1 2.7 50 5 0 5.1 50 5 0 2.7 50 5 0 5.1 50 5 0 2.7 50 5 0 5.1 50 5 0 2.7 50 5 0 5.1 50 5 0 2.7 50 5 0 5.1 50 5 0 2.7 50 10 0 5.1 50 10 0 2.7 50 10 0 5.1 50 10 0 2.7 50 10 0 5.1 50 10 0 2.7 50 10 1 5.1 50 10 0 2.7 50 10 1 5.1 50 10 0 2.7 50 15 1 5.1 50 15 0 2.7 50 15 1 5.1 50 15 0 2.7 50 15 1 5.1 50 15 0 2.7 50 15 1 5.1 50 15 1 2.7 50 15 1 5.1 50 15 1 2.7 50 20 1 5.1 50 20 1 2.7 50 20 1 5.1 50 20 1 2.7 50 20 1 5.1 50 20 1 2.7 50 20 1 5.1 50 20 1 2.7 50 20 1 5.1 50 20 1 2.7 50 25 1 5.1 50 25 1 2.7 50 25 1 5.1 50 25 1 2.7 50 25 1 5.1 50 25 1 2.7 50 25 1 5.1 50 25 1 2.7 50 25 1 5.1 50 25 1 2.7 50 30 1 5.1 50 30 1 2.7 50 30 1 5.1 50 30 1 2.7 50 30 1 5.1 50 30 1 2.7 50 30 1 5.1 50 30 1 2.7 50 30 1 5.1 50 30 1 2.7 75 5 0 5.1 75 5 0 2.7 75 5 0 5.1 75 5 0 2.7 75 5 1 5.1 75 5 0 2.7 75 5 1 5.1 75 5 0 2.7 75 5 1 5.1 75 5 1 2.7 75 10 1 5.1 75 10 0 2.7 75 10 1 5.1 75 10 0 2.7 75 10 1 5.1 75 10 1 2.7 75 10 1 5.1 75 10 1 2.7 75 10 1 5.1 75 10 1 2.7 75 15 1 5.1 75 15 1 2.7 75 15 1 5.1 75 15 1 2.7 75 15 1 5.1 75 15 1 2.7 75 15 1 5.1 75 15 1 2.7 75 15 1 5.1 75 15 1 2.7 75 20 1 5.1 75 20 1 2.7 75 20 1 5.1 75 20 1 2.7 75 20 1 5.1 75 20 1 2.7 75 20 1 5.1 75 20 1 2.7 75 20 1 5.1 75 20 1 2.7 75 25 1 5.1 75 25 1 2.7 75 25 1 5.1 75 25 1 2.7 75 25 1 5.1 75 25 1 2.7 75 25 1 5.1 75 25 1 2.7 75 25 1 5.1 75 25 1 2.7 75 30 1 5.1 75 30 1 2.7 75 30 1 5.1 75 30 1 2.7 75 30 1 5.1 75 30 1 2.7 75 30 1 5.1 75 30 1 2.7 75 30 1 5.1 75 30 1 3.4 25 5 0 9 25 5 0 3.4 25 5 0 9 25 5 0 3.4 25 5 0 9 25 5 0 3.4 25 5 0 9 25 5 0 3.4 25 5 0 9 25 5 0 3.4 25 10 0 9 25 10 0 3.4 25 10 0 9 25 10 0 3.4 25 10 0 9 25 10 0 3.4 25 10 0 9 25 10 0 3.4 25 10 0 9 25 10 0 3.4 25 15 0 9 25 15 0 3.4 25 15 0 9 25 15 0 3.4 25 15 0 9 25 15 0 3.4 25 15 1 9 25 15 0 3.4 25 15 1 9 25 15 0 3.4 25 20 0 9 25 20 0 3.4 25 20 1 9 25 20 0 3.4 25 20 1 9 25 20 0 3.4 25 20 1 9 25 20 0 3.4 25 20 1 9 25 20 0 3.4 25 25 1 9 25 25 0 3.4 25 25 1 9 25 25 0 3.4 25 25 1 9 25 25 0 3.4 25 25 1 9 25 25 0 3.4 25 25 1 9 25 25 0 3.4 25 30 1 9 25 30 0 3.4 25 30 1 9 25 30 0 3.4 25 30 1 9 25 30 0 3.4 25 30 1 9 25 30 1 3.4 25 30 1 9 25 30 1 3.4 50 5 0 9 50 5 0 3.4 50 5 0 9 50 5 0 3.4 50 5 0 9 50 5 0 3.4 50 5 0 9 50 5 0 3.4 50 5 0 9 50 5 0 3.4 50 10 1 9 50 10 0 3.4 50 10 1 9 50 10 0 3.4 50 10 1 9 50 10 0 3.4 50 10 1 9 50 10 0 3.4 50 10 1 9 50 10 0 3.4 50 15 1 9 50 15 0 3.4 50 15 1 9 50 15 0 3.4 50 15 1 9 50 15 0 3.4 50 15 1 9 50 15 0 3.4 50 15 1 9 50 15 1 3.4 50 20 1 9 50 20 0 3.4 50 20 1 9 50 20 0 3.4 50 20 1 9 50 20 0 3.4 50 20 1 9 50 20 1 3.4 50 20 1 9 50 20 1 3.4 50 25 1 9 50 25 0 3.4 50 25 1 9 50 25 1 3.4 50 25 1 9 50 25 1 3.4 50 25 1 9 50 25 1 3.4 50 25 1 9 50 25 1 3.4 50 30 1 9 50 30 1 3.4 50 30 1 9 50 30 1 3.4 50 30 1 9 50 30 1 3.4 50 30 1 9 50 30 1 3.4 50 30 1 9 50 30 1 3.4 75 5 0 9 75 5 0 3.4 75 5 0 9 75 5 0 3.4 75 5 1 9 75 5 0 3.4 75 5 1 9 75 5 1 3.4 75 5 1 9 75 5 1 3.4 75 10 1 9 75 10 0 3.4 75 10 1 9 75 10 1 3.4 75 10 1 9 75 10 1 3.4 75 10 1 9 75 10 1 3.4 75 10 1 9 75 10 1 3.4 75 15 1 9 75 15 1 3.4 75 15 1 9 75 15 1 3.4 75 15 1 9 75 15 1 3.4 75 15 1 9 75 15 1 3.4 75 15 1 9 75 15 1 3.4 75 20 1 9 75 20 1 3.4 75 20 1 9 75 20 1 3.4 75 20 1 9 75 20 1 3.4 75 20 1 9 75 20 1 3.4 75 20 1 9 75 20 1 3.4 75 25 1 9 75 25 1 3.4 75 25 1 9 75 25 1 3.4 75 25 1 9 75 25 1 3.4 75 25 1 9 75 25 1 3.4 75 25 1 9 75 25 1 3.4 75 30 1 9 75 30 1 3.4 75 30 1 9 75 30 1 3.4 75 30 1 9 75 30 1 3.4 75 30 1 9 75 30 1 3.4 75 30 1 9 75 30 1

TABLE 5A 2 Data pertaining to MPS models with cross section of about 25 cm Length Length BS BP SCS (L) cm BS BP SCS (L) cm 2.7 25 25 13 5.1 25 25 38 4 25 25 15 5.1 50 25 39 9 25 25 16 2.6 25 25 40 9 25 25 18 5.1 75 25 44 4 25 25 18 2.7 25 25 44 9 50 25 19 2.7 50 25 45 5.1 50 25 19 3.4 50 25 46 9 75 25 23 5.1 75 25 49 2.6 25 25 23 4 50 25 49 9 50 25 24 5.1 25 25 51 4 25 25 24 2.6 50 25 52 3.4 50 25 24 4 75 25 52 2.6 25 25 24 3.4 50 25 56 9 50 25 25 2.6 25 25 57 5.1 25 25 25 5.1 75 25 58 4 50 25 25 4 50 25 58 2.7 25 25 27 4 75 25 60 5.1 25 25 27 9 50 25 61 4 25 25 28 9 25 25 62 3.4 25 25 28 2.7 75 25 65 2.6 25 25 28 2.7 25 25 66 3.4 25 25 32 2.6 50 25 66 3.4 25 25 32 2.6 50 25 68 5.1 50 25 33 3.4 25 25 69 4 50 25 33 9 75 25 71 5.1 25 25 33 2.7 25 25 71 9 25 25 34 5.1 75 25 72 9 75 25 35 3.4 25 25 72 9 25 25 35 2.6 75 25 75 4 25 25 35 5.1 75 25 77 2.6 50 25 35 2.6 50 25 79 4 50 25 36 2.6 75 25 80 3.4 50 25 36 4 75 25 90 5.1 50 25 36 4 75 25 92 9 50 25 37 2.7 75 25 92 5.1 50 25 92

TABLE 5B Probability of having LCBV represented by 5 cm long samples (pLCBV) pLCBV = pLCBV = (L/5) − ((L/5) − (L/5) − ((L/5) − L L/5 2 2)/20 L L/5 2 2)/20 13 2.6 0.6 0.03 39 7.8 5.8 0.29 15 3 1 0.05 40 8 6 0.3 16 3.2 1.2 0.06 44 8.8 6.8 0.34 18 3.6 1.6 0.08 44 8.8 6.8 0.34 18 3.6 1.6 0.08 45 9 7 0.35 19 3.8 1.8 0.09 46 9.2 7.2 0.36 19 3.8 1.8 0.09 49 9.8 7.8 0.39 23 4.6 2.6 0.13 49 9.8 7.8 0.39 23 4.6 2.6 0.13 51 10.2 8.2 0.41 24 4.8 2.8 0.14 52 10.4 8.4 0.42 24 4.8 2.8 0.14 52 10.4 8.4 0.42 24 4.8 2.8 0.14 56 11.2 9.2 0.46 24 4.8 2.8 0.14 57 11.4 9.4 0.47 25 5 3 0.15 58 11.6 9.6 0.48 25 5 3 0.15 58 11.6 9.6 0.48 25 5 3 0.15 60 12 10 0.5 27 5.4 3.4 0.17 61 12.2 10.2 0.51 27 5.4 3.4 0.17 62 12.4 10.4 0.52 28 5.6 3.6 0.18 65 13 11 0.55 28 5.6 3.6 0.18 66 13.2 11.2 0.56 28 5.6 3.6 0.18 66 13.2 11.2 0.56 32 6.4 4.4 0.22 68 13.6 11.6 0.58 32 6.4 4.4 0.22 69 13.8 11.8 0.59 33 6.6 4.6 0.23 71 14.2 12.2 0.61 33 6.6 4.6 0.23 71 14.2 12.2 0.61 33 6.6 4.6 0.23 72 14.4 12.4 0.62 34 6.8 4.8 0.24 72 14.4 12.4 0.62 35 7 5 0.25 75 15 13 0.65 35 7 5 0.25 77 15.4 13.4 0.67 35 7 5 0.25 79 15.8 13.8 0.69 35 7 5 0.25 80 16 14 0.7 36 7.2 5.2 0.26 90 18 16 0.8 36 7.2 5.2 0.26 92 18.4 16.4 0.82 36 7.2 5.2 0.26 92 18.4 16.4 0.82 37 7.4 5.4 0.27 92 18.4 16.4 0.82 38 7.6 5.6 0.28

TABLE 5C Probability of having LCBV represented by 2.5 cm long samples (pLCBV) pLCBV = pLCBV = (L/5) − ((L/5) − (L/5) − ((L/5) − L L/2.5 2 2)/40 L L/2.5 2 2)/40 13 5.2 3.2 0.08 39 7.8 5.8 0.29 15 6 4 0.1 40 8 6 0.3 16 6.4 4.4 0.11 44 8.8 6.8 0.34 18 7.2 5.2 0.13 44 8.8 6.8 0.34 18 7.2 5.2 0.13 45 9 7 0.35 19 7.6 5.6 0.14 46 9.2 7.2 0.36 19 7.6 5.6 0.14 49 9.8 7.8 0.39 23 9.2 7.2 0.18 49 9.8 7.8 0.39 23 9.2 7.2 0.18 51 10.2 8.2 0.41 24 9.6 7.6 0.19 52 10.4 8.4 0.42 24 9.6 7.6 0.19 52 10.4 8.4 0.42 24 9.6 7.6 0.19 56 11.2 9.2 0.46 24 9.6 7.6 0.19 57 11.4 9.4 0.47 25 10 8 0.2 58 11.6 9.6 0.48 25 10 8 0.2 58 11.6 9.6 0.48 25 10 8 0.2 60 12 10 0.5 27 10.8 8.8 0.22 61 12.2 10.2 0.51 27 10.8 8.8 0.22 62 12.4 10.4 0.52 28 11.2 9.2 0.23 65 13 11 0.55 28 11.2 9.2 0.23 66 13.2 11.2 0.56 28 11.2 9.2 0.23 66 13.2 11.2 0.56 32 12.8 10.8 0.27 68 13.6 11.6 0.58 32 12.8 10.8 0.27 69 13.8 11.8 0.59 33 13.2 11.2 0.28 71 14.2 12.2 0.61 33 13.2 11.2 0.28 71 14.2 12.2 0.61 33 13.2 11.2 0.28 72 14.4 12.4 0.62 34 13.6 11.6 0.29 72 14.4 12.4 0.62 35 14 12 0.3 75 15 13 0.65 35 14 12 0.3 77 15.4 13.4 0.67 35 14 12 0.3 79 15.8 13.8 0.69 35 14 12 0.3 80 16 14 0.7 36 14.4 12.4 0.31 90 18 16 0.8 36 14.4 12.4 0.31 92 18.4 16.4 0.82 36 14.4 12.4 0.31 92 18.4 16.4 0.82 37 14.8 12.8 0.32 92 18.4 16.4 0.82 38 15.2 13.2 0.33

TABLE 5D Probability of having LCBV represented by 1 cm long samples (pLCBV) pLCBV = pLCBV = (L/5) − ((L/5) − (L/5) − ((L/5) − L L/1 2 2)/100 L L/1 2 2)/100 13 13 11 0.11 39 39 37 0.37 15 15 13 0.13 40 40 38 0.38 16 16 14 0.14 44 44 42 0.42 18 18 16 0.16 44 44 42 0.42 18 18 16 0.16 45 45 43 0.43 19 19 17 0.17 46 46 44 0.44 19 19 17 0.17 49 49 47 0.47 23 23 21 0.21 49 49 47 0.47 23 23 21 0.21 51 51 49 0.49 24 24 22 0.22 52 52 50 0.5 24 24 22 0.22 52 52 50 0.5 24 24 22 0.22 56 56 54 0.54 24 24 22 0.22 57 57 55 0.55 25 25 23 0.23 58 58 56 0.56 25 25 23 0.23 58 58 56 0.56 25 25 23 0.23 60 60 58 0.58 27 27 25 0.25 61 61 59 0.59 27 27 25 0.25 62 62 60 0.6 28 28 26 0.26 65 65 63 0.63 28 28 26 0.26 66 66 64 0.64 28 28 26 0.26 66 66 64 0.64 32 32 30 0.3 68 68 66 0.66 32 32 30 0.3 69 69 67 0.67 33 33 31 0.31 71 71 69 0.69 33 33 31 0.31 71 71 69 0.69 33 33 31 0.31 72 72 70 0.7 34 34 32 0.32 72 72 70 0.7 35 35 33 0.33 75 75 73 0.73 35 35 33 0.33 77 77 75 0.75 35 35 33 0.33 79 79 77 0.77 35 35 33 0.33 80 80 78 0.78 36 36 34 0.34 90 90 88 0.88 36 36 34 0.34 92 92 90 0.9 36 36 34 0.34 92 92 90 0.9 37 37 35 0.35 92 92 90 0.9 38 38 36 0.36

TABLE 6A 2 Data pertaining to MPS models with cross section of 100 cm BS BP SCS Length (L) cm 9 25 100 16 9 25 100 19 5.1 25 100 24 9 50 100 25 5.1 25 100 26 5.1 50 100 32 9 25 100 34 5.1 50 100 34 9 25 100 36 4 25 100 37 5.1 50 100 42 4 25 100 42 5.1 25 100 43 4 25 100 47 5.1 25 100 48 4 25 100 49 3.4 25 100 49 3.4 25 100 49 2.7 25 100 50 5.1 50 100 53 9 25 100 54 4 50 100 60 4 50 100 60 5.1 25 100 64 4 50 100 64 3.4 25 100 64 2.7 50 100 64 2.6 25 100 67 9 50 100 72 9 50 100 73 4 50 100 74 2.7 50 100 76 2.7 25 100 80 2.6 25 100 80 5.1 50 100 84 5.1 75 100 84 9 50 100 85 9 50 100 90 2.6 25 100 91 3.4 25 100 96 2.7 25 100 96 2.7 25 100 98

TABLE 6B Probability of having LCBV represented by 10 cm long samples (pLCBV) L L/5 (L/5) − 2 pLCBV = ((L/5) − 2)/10 16 1.6 −0.4 −0.04 19 1.9 −0.1 −0.01 24 2.4 0.4 0.04 25 2.5 0.5 0.05 26 2.6 0.6 0.06 32 3.2 1.2 0.12 34 3.4 1.4 0.14 34 3.4 1.4 0.14 36 3.6 1.6 0.16 37 3.7 1.7 0.17 42 4.2 2.2 0.22 42 4.2 2.2 0.22 43 4.3 2.3 0.23 47 4.7 2.7 0.27 48 4.8 2.8 0.28 49 4.9 2.9 0.29 49 4.9 2.9 0.29 49 4.9 2.9 0.29 50 5 3 0.3 53 5.3 3.3 0.33 54 5.4 3.4 0.34 60 6 4 0.4 60 6 4 0.4 64 6.4 4.4 0.44 64 6.4 4.4 0.44 64 6.4 4.4 0.44 64 6.4 4.4 0.44 67 6.7 4.7 0.47 72 7.2 5.2 0.52 73 7.3 5.3 0.53 74 7.4 5.4 0.54 76 7.6 5.6 0.56 80 8 6 0.6 80 8 6 0.6 84 8.4 6.4 0.64 84 8.4 6.4 0.64 85 8.5 6.5 0.65 90 9 7 0.7 91 9.1 7.1 0.71 96 9.6 7.6 0.76 96 9.6 7.6 0.76 98 9.8 7.8 0.78

TABLE 6C Probability of having LCBV represented by 20 cm long samples (pLCBV) L L/2.5 (L/5) − 2 pLCBV = ((L/5) − 2)/5 16 0.8 −1.2 −0.24 19 0.95 −1.05 −0.21 24 1.2 −0.8 −0.16 25 1.25 −0.75 −0.15 26 1.3 −0.7 −0.14 32 1.6 −0.4 −0.08 34 1.7 −0.3 −0.06 34 1.7 −0.3 −0.06 36 1.8 −0.2 −0.04 37 1.85 −0.15 −0.03 42 2.1 0.1 0.02 42 2.1 0.1 0.02 43 2.15 0.15 0.03 47 2.35 0.35 0.07 48 2.4 0.4 0.08 49 2.45 0.45 0.09 49 2.45 0.45 0.09 49 2.45 0.45 0.09 50 2.5 0.5 0.1 53 2.65 0.65 0.13 54 2.7 0.7 0.14 60 3 1 0.2 60 3 1 0.2 64 3.2 1.2 0.24 64 3.2 1.2 0.24 64 3.2 1.2 0.24 64 3.2 1.2 0.24 67 3.35 1.35 0.27 72 3.6 1.6 0.32 73 3.65 1.65 0.33 74 3.7 1.7 0.34 76 3.8 1.8 0.36 80 4 2 0.4 80 4 2 0.4 84 4.2 2.2 0.44 84 4.2 2.2 0.44 85 4.25 2.25 0.45 90 4.5 2.5 0.5 91 4.55 2.55 0.51 96 4.8 2.8 0.56 96 4.8 2.8 0.56 98 4.9 2.9 0.58

TABLE 6D Probability of having LCBV represented by 25 cm long samples (pLCBV) L L/1 (L/5) − 2 pLCBV = ((L/5) − 2)/4 16 0.64 −1.36 −0.34 19 0.76 −1.24 −0.31 24 0.96 −1.04 −0.26 25 1 −1 −0.25 26 1.04 −0.96 −0.24 32 1.28 −0.72 −0.18 34 1.36 −0.64 −0.16 34 1.36 −0.64 −0.16 36 1.44 −0.56 −0.14 37 1.48 −0.52 −0.13 42 1.68 −0.32 −0.08 42 1.68 −0.32 −0.08 43 1.72 −0.28 −0.07 47 1.88 −0.12 −0.03 48 1.92 −0.08 −0.02 49 1.96 −0.04 −0.01 49 1.96 −0.04 −0.01 49 1.96 −0.04 −0.01 50 2 0 0 53 2.12 0.12 0.03 54 2.16 0.16 0.04 60 2.4 0.4 0.1 60 2.4 0.4 0.1 64 2.56 0.56 0.14 64 2.56 0.56 0.14 64 2.56 0.56 0.14 64 2.56 0.56 0.14 67 2.68 0.68 0.17 72 2.88 0.88 0.22 73 2.92 0.92 0.23 74 2.96 0.96 0.24 76 3.04 1.04 0.26 80 3.2 1.2 0.3 80 3.2 1.2 0.3 84 3.36 1.36 0.34 84 3.36 1.36 0.34 85 3.4 1.4 0.35 90 3.6 1.6 0.4 91 3.64 1.64 0.41 96 3.84 1.84 0.46 96 3.84 1.84 0.46 98 3.92 1.92 0.48

7 7 FIGS.A-F 8 8 FIGS.A-C 9 9 FIGS.A-C 10 FIG. 10 FIG. 100 200 100 2 2 2 The results highlighted the importance of BP, BS and SCS as variables that control the representation of the Thalassinoides connectivity in samples, as shown in,, and. Further, it may be quantified from the above stated examples that BP, BS and SCS quantitatively control representation of Thalassinoides connectivity when sampling. The systemand the methodprovided in the present disclosure are vital to determine the correct dimension of samples that can represent Thalassinoides connectivity and acquiring permeability measurements to represent a reservoir volume containing Thalassinoides. The logistic regression model developed in the present disclosure provides a method to collectively link BP, BS and SCS as controls on Thalassinoides connectivity across a 1-meter-long sample. Furthermore, the weight of each of these variables as controllers on the Thalassinoides connectivity is calculated by the logistic regression model, as shown in Tables 3A-3I. An important product of the logistic regression model is the equation of the probability of representing Thalassinoides connectivity in 1-meter-long digital samples. Where the logistic regression produces a high probability for an SCS to show Thalassinoides connectivity across that 1 m sample, a shorter segment of that same 1 m sample may show the same connectivity. Thus, the equation provided by the present disclosure can be used as a lower limit on the SCS required to represent Thalassinoides permeability. In one implementation of the present disclosure, to measure permeability on a sample that is shorter than 1 m, the BP and BS of that sample may be fed as input to the systemand further calculate the SCS having the desired high probability of connecting across the 1 m sample, an example is documented in Table 7. In addition, the output SCS may be used for sampling and measuring. In another implementation of the present disclosure, the Thalassinoides-bearing strata in the Hanifa formation in central Saudi Arabia was examined. Thalassinoides of the Hanifa formation were examined and burrow attributes, including BP (range from 10% to 40%) and BS (shaft diameters range from 0.86 cm to 3.6 cm with mean of 1.5 cm), were documented. A particularly large diameter core (drilled with a 10-inch bit) was studied to characterize Thalassinoides using CT-scans. The core was characterized in its entirety and evaluated to examine the scale of subsamples that would produce connectivity across the entire 26.5 cm length of the core. The CT scan of the full sample showed a BP of 38%. The Thalassinoides in that large core connected across the top and bottom of the core. In one case, if 1.5 cm BS and 38% BP is provided as input to the logistic regression equation developed herein, it produces a probability curve of Thalassinoides connecting across a 1 m-long sample given various sample cross sections, as shown in. Probabilities greater than 0.5 require sample cross sections of 64 cm(8 cm side length) and higher, as shown in. By digitally subsampling the scanned 26.5 cm-long core, it was found that prisms with cross sections of 51.84 cm(7.2 cm side length) showed 100% probability of Thalassinoides connecting across the top and bottom, and samples with cross sections of about 30 cm(5.5 cm side length) showed 50% probability of Thalassinoides connecting. The SCS output from the 1-m logistic regression equation is slightly larger than the SCS required for Thalassinoides connectivity in the 26.5 cm example. This supports the results from the logistic regression equation, and can be used to guide the minimum SCS of samples that are shorter that 1 m, sample lengths within the realm of easy analysis by standard techniques.

TABLE 7 Probability of Connectivity (PC) of 1-meter-long columnar sample PC BS (cm) PB (%) 2 SCS (cm) 0.13 1.5 25 1 0.134 1.5 25 4 0.139 1.5 25 9 0.148 1.5 25 16 0.159 1.5 25 25 0.174 1.5 25 36 0.193 1.5 25 49 0.216 1.5 25 64 0.245 1.5 25 81 0.281 1.5 25 100 0.323 1.5 25 121 0.374 1.5 25 144 0.432 1.5 25 169 0.496 1.5 25 196 0.566 1.5 25 225 0.638 1.5 25 256 0.708 1.5 25 289 0.772 1.5 25 324 0.829 1.5 25 361 0.876 1.5 25 400 0.913 1.5 25 441 0.941 1.5 25 484 0.961 1.5 25 529 0.975 1.5 25 576 0.984 1.5 25 625 0.99 1.5 25 676 0.994 1.5 25 729 0.997 1.5 25 784 0.998 1.5 25 841 0.999 1.5 25 900

13 FIG. 13 FIG. 1 FIG. 1300 100 1301 1302 1304 Next, further details of the hardware description of the computing environment according to exemplary embodiments is described with reference to. In, a controlleris described is representative of the systemofin which the controller is a computing device which includes a CPUwhich performs the processes described above/below. The process data and instructions may be stored in memory. These processes and instructions may also be stored on a storage medium disksuch as a hard drive (HDD) or portable storage medium or may be stored remotely.

Further, the claims are not limited by the form of the computer-readable media on which the instructions of the inventive process are stored. For example, the instructions may be stored on CDs, DVDs, in FLASH memory, RAM, ROM, PROM, EPROM, EEPROM, hard disk or any other information processing device with which the computing device communicates, such as a server or computer.

1301 1303 Further, the claims may be provided as a utility application, background daemon, or component of an operating system, or combination thereof, executing in conjunction with CPU,and an operating system such as Microsoft Windows 7, Microsoft Windows 10, Microsoft Windows 11, UNIX, Solaris, LINUX, Apple MAC-OS and other systems known to those skilled in the art.

1301 1303 1301 1303 1301 1303 The hardware elements in order to achieve the computing device may be realized by various circuitry elements, known to those skilled in the art. For example, CPUor CPUmay be a Xenon or Core processor from Intel of America or an Opteron processor from AMD of America, or may be other processor types that would be recognized by one of ordinary skill in the art. Alternatively, the CPU,may be implemented on an FPGA, ASIC, PLD or using discrete logic circuits, as one of ordinary skill in the art would recognize. Further, CPU,may be implemented as multiple processors cooperatively working in parallel to perform the instructions of the inventive processes described above.

13 FIG. 1306 1360 1360 1360 The computing device inalso includes a network controller, such as an Intel Ethernet PRO network interface card from Intel Corporation of America, for interfacing with network. As can be appreciated, the networkcan be a public network, such as the Internet, or a private network such as an LAN or WAN network, or any combination thereof and can also include PSTN or ISDN sub-networks. The networkcan also be wired, such as an Ethernet network, or can be wireless such as a cellular network including EDGE, 3G, 4G and 5G wireless cellular systems. The wireless network can also be Wi-Fi, Bluetooth, or any other wireless form of communication that is known.

1308 1310 1312 1314 1316 1310 1318 The computing device further includes a display controller, such as a NVIDIA GeForce GTX or Quadro graphics adaptor from NVIDIA Corporation of America for interfacing with display, such as a Hewlett Packard HPL2445w LCD monitor. A general purpose I/O interfaceinterfaces with a keyboard and/or mouseas well as a touch screen panelon or separate from display. General purpose I/O interface also connects to a variety of peripheralsincluding printers and scanners, such as an OfficeJet or DeskJet from Hewlett Packard.

1320 1322 A sound controlleris also provided in the computing device such as Sound Blaster X-Fi Titanium from Creative, to interface with speakers/microphonethereby providing sounds and/or music.

1324 1304 1326 1310 1314 1308 1324 1306 1320 1312 The general-purpose storage controllerconnects the storage medium diskwith communication bus, which may be an ISA, EISA, VESA, PCI, or similar, for interconnecting all of the components of the computing device. A description of the general features and functionality of the display, keyboard and/or mouse, as well as the display controller, storage controller, network controller, sound controller, and general purpose I/O interfaceis omitted herein for brevity as these features are known.

14 FIG. The exemplary circuit elements described in the context of the present disclosure may be replaced with other elements and structured differently than the examples provided herein. Moreover, circuitry configured to perform features described herein may be implemented in multiple circuit units (e.g., chips), or the features may be combined in circuitry on a single chipset, as shown on.

14 FIG. shows a schematic diagram of a data processing system, according to certain embodiments, for performing the functions of the exemplary embodiments. The data processing system is an example of a computer in which code or instructions implementing the processes of the illustrative embodiments may be located.

4 FIG. 1400 1425 1420 1430 1425 1425 1445 1450 1425 1420 1430 In, data processing systememploys a hub architecture including a north bridge and memory controller hub (NB/MCH)and a south bridge and input/output (I/O) controller hub (SB/ICH). The central processing unit (CPU)is connected to NB/MCH. The NB/MCHalso connects to the memoryvia a memory bus and connects to the graphics processorvia an accelerated graphics port (AGP). The NB/MCHalso connects to the SB/ICHvia an internal bus (e.g., a unified media interface or a direct media interface). The CPU Processing unitmay contain one or more processors and even may be implemented using one or more heterogeneous processor systems.

15 FIG. 1430 1538 1540 1538 1536 1430 1532 1534 1532 1540 1430 1430 1430 1430 For example,shows one implementation of CPU. In one implementation, the instruction registersretrieves instructions from the fast memory. At least part of these instructions is fetched from the instruction registerby the control logicand interpreted according to the instruction set architecture of the CPU. Part of the instructions can also be directed to the register. In one implementation the instructions are decoded according to a hardwired method, and in another implementation the instructions are decoded according to a microprogram that translates instructions into sets of CPU configuration signals that are applied sequentially over multiple clock pulses. After fetching and decoding the instructions, the instructions are executed using the arithmetic logic unit (ALU)that loads values from the registerand performs logical and mathematical operations on the loaded values according to the instructions. The results from these operations can be feedback into the register and/or stored in the fast memory. According to certain implementations, the instruction set architecture of the CPUcan use a reduced instruction set architecture, a complex instruction set architecture, a vector processor architecture, a very large instruction word architecture. Furthermore, the CPUcan be based on the Von Neuman model or the Harvard model. The CPUcan be a digital signal processor, an FPGA, an ASIC, a PLA, a PLD, or a CPLD. Further, the CPUcan be an x86 processor by Intel or by AMD; an ARM processor, a Power architecture processor by, e.g., IBM; a SPARC architecture processor by Sun Microsystems or by Oracle; or other known CPU architecture.

14 FIG. 1400 1420 1456 1464 1468 1458 1488 1462 Referring again to, the data processing systemcan include that the SB/ICHis coupled through a system bus to an I/O Bus, a read only memory (ROM), universal serial bus (USB) port, a flash binary input/output system (BIOS), and a graphics controller. PCI/PCIe devices can also be coupled to SB/ICHthrough a PCI bus.

1460 1466 The PCI devices may include, for example, Ethernet adapters, add-in cards, and PC cards for notebook computers. The Hard disk driveand CD-ROMcan use, for example, an integrated drive electronics (IDE) or serial advanced technology attachment (SATA) interface. In one implementation the I/O bus can include a super I/O (SIO) device.

1460 1466 1420 1470 1472 1478 1476 1420 Further, the hard disk drive (HDD)and optical drivecan also be coupled to the SB/ICHthrough a system bus. In one implementation, a keyboard, a mouse, a parallel port, and a serial portcan be connected to the system bus through the I/O bus. Other peripherals and devices that can be connected to the SB/ICHusing a mass storage controller such as SATA or PATA, an Ethernet port, an ISA bus, a LPC bridge, SMBus, a DMA controller, and an Audio Codec.

Moreover, the present disclosure is not limited to the specific circuit elements described herein, nor is the present disclosure limited to the specific sizing and classification of these elements. For example, the skilled artisan will appreciate that the circuitry described herein may be adapted based on changes on battery sizing and chemistry or based on the requirements of the intended back-up load to be powered.

1630 1636 1632 1634 1638 1640 1620 1622 1624 1626 1616 1610 1612 1614 1652 1654 16 FIG. The functions and features described herein may also be executed by various distributed components of a system. For example, one or more processors may execute these system functions, wherein the processors are distributed across multiple components communicating in a network. The distributed components may include one or more client and server machines, such as cloudincluding a cloud controller, a secure gateway, a data centre, data storageand a provisioning tool, and mobile network servicesincluding central processors, a serverand a database, which may share processing, as shown by, in addition to various human interface and communication devices (e.g., display monitors, smart phones, tablets, personal digital assistants (PDAs)). The network may be a private network, such as a LAN, satelliteor WAN, or be a public network, may such as the Internet. Input to the system may be received via direct user input and received remotely either in real-time or as a batch process. Additionally, some implementations may be performed on modules or hardware not identical to those described. Accordingly, other implementations are within the scope that may be claimed.

The above-described hardware description is a non-limiting example of corresponding structure for performing the functionality described herein.

Numerous modifications and variations of the present disclosure are possible in light of the above teachings. It is therefore to be understood that within the scope of the appended claims, the invention may be practiced otherwise than as specifically described herein.